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Theorem updjud 9870
Description: Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9838 and df-inr 9839, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9839 (25-Aug-2023). This is a special case of Example 1 of coproducts in Section 10.67 of [Adamek] p. 185. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
Hypotheses
Ref Expression
updjud.f (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
updjud.g (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
updjud.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
updjud.b (πœ‘ β†’ 𝐡 ∈ π‘Š)
Assertion
Ref Expression
updjud (πœ‘ β†’ βˆƒ!β„Ž(β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺))
Distinct variable groups:   𝐴,β„Ž   𝐡,β„Ž   𝐢,β„Ž   β„Ž,𝐹   β„Ž,𝐺   πœ‘,β„Ž
Allowed substitution hints:   𝑉(β„Ž)   π‘Š(β„Ž)

Proof of Theorem updjud
Dummy variables π‘˜ π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 updjud.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑉)
2 updjud.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ π‘Š)
31, 2jca 512 . . . . 5 (πœ‘ β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š))
4 djuex 9844 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 βŠ” 𝐡) ∈ V)
5 mptexg 7171 . . . . 5 ((𝐴 βŠ” 𝐡) ∈ V β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∈ V)
63, 4, 53syl 18 . . . 4 (πœ‘ β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∈ V)
7 feq1 6649 . . . . . . 7 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ↔ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢))
8 coeq1 5813 . . . . . . . 8 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (β„Ž ∘ (inl β†Ύ 𝐴)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)))
98eqeq1d 2738 . . . . . . 7 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ ((β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ↔ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹))
10 coeq1 5813 . . . . . . . 8 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (β„Ž ∘ (inr β†Ύ 𝐡)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)))
1110eqeq1d 2738 . . . . . . 7 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ ((β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺 ↔ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺))
127, 9, 113anbi123d 1436 . . . . . 6 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ ((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ↔ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)))
13 eqeq1 2740 . . . . . . . 8 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (β„Ž = π‘˜ ↔ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))
1413imbi2d 340 . . . . . . 7 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜) ↔ ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜)))
1514ralbidv 3174 . . . . . 6 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜) ↔ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜)))
1612, 15anbi12d 631 . . . . 5 (β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) β†’ (((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜)) ↔ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))))
1716adantl 482 . . . 4 ((πœ‘ ∧ β„Ž = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))) β†’ (((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜)) ↔ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))))
18 updjud.f . . . . . 6 (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
19 updjud.g . . . . . 6 (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
20 eqid 2736 . . . . . 6 (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))
2118, 19, 20updjudhf 9867 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢)
2218, 19, 20updjudhcoinlf 9868 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹)
2318, 19, 20updjudhcoinrg 9869 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)
24 simpr 485 . . . . . . 7 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺))
25 eqeq2 2748 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) ↔ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹))
26 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ 𝐴 β†’ ((inl β†Ύ 𝐴)β€˜π‘§) = (inlβ€˜π‘§))
2726eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ 𝐴 β†’ (inlβ€˜π‘§) = ((inl β†Ύ 𝐴)β€˜π‘§))
2827eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ 𝐴 β†’ (𝑦 = (inlβ€˜π‘§) ↔ 𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§)))
2928adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ (𝑦 = (inlβ€˜π‘§) ↔ 𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§)))
30 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴))β€˜π‘§) = ((π‘˜ ∘ (inl β†Ύ 𝐴))β€˜π‘§))
3130ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴))β€˜π‘§) = ((π‘˜ ∘ (inl β†Ύ 𝐴))β€˜π‘§))
32 inlresf 9850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)
33 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡) β†’ (inl β†Ύ 𝐴) Fn 𝐴)
3432, 33mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) β†’ (inl β†Ύ 𝐴) Fn 𝐴)
35 fvco2 6938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl β†Ύ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴))β€˜π‘§) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
3634, 35sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴))β€˜π‘§) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
37 fvco2 6938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl β†Ύ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴))β€˜π‘§) = (π‘˜β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
3834, 37sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴))β€˜π‘§) = (π‘˜β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
3931, 36, 383eqtr3d 2784 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inl β†Ύ 𝐴)β€˜π‘§)) = (π‘˜β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
40 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
41 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§) β†’ (π‘˜β€˜π‘¦) = (π‘˜β€˜((inl β†Ύ 𝐴)β€˜π‘§)))
4240, 41eqeq12d 2752 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦) ↔ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inl β†Ύ 𝐴)β€˜π‘§)) = (π‘˜β€˜((inl β†Ύ 𝐴)β€˜π‘§))))
4339, 42syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ (𝑦 = ((inl β†Ύ 𝐴)β€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
4429, 43sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐴) β†’ (𝑦 = (inlβ€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
4544expimpd 454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) ∧ πœ‘) β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
4645ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)) β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
4746eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘˜ ∘ (inl β†Ύ 𝐴)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
4825, 47syl6bir 253 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
4948com23 86 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ (πœ‘ β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
50493ad2ant2 1134 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (πœ‘ β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
5150impcom 408 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
5251com12 32 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 β†’ ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
53523ad2ant2 1134 . . . . . . . . . . . . . . . 16 ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
5453impcom 408 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
5554com12 32 . . . . . . . . . . . . . 14 ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘§)) β†’ (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
5655rexlimiva 3144 . . . . . . . . . . . . 13 (βˆƒπ‘§ ∈ 𝐴 𝑦 = (inlβ€˜π‘§) β†’ (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
57 eqeq2 2748 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) ↔ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺))
58 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ 𝐡 β†’ ((inr β†Ύ 𝐡)β€˜π‘§) = (inrβ€˜π‘§))
5958eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ 𝐡 β†’ (inrβ€˜π‘§) = ((inr β†Ύ 𝐡)β€˜π‘§))
6059eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ 𝐡 β†’ (𝑦 = (inrβ€˜π‘§) ↔ 𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§)))
6160adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ (𝑦 = (inrβ€˜π‘§) ↔ 𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§)))
62 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡))β€˜π‘§) = ((π‘˜ ∘ (inr β†Ύ 𝐡))β€˜π‘§))
6362ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡))β€˜π‘§) = ((π‘˜ ∘ (inr β†Ύ 𝐡))β€˜π‘§))
64 inrresf 9852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inr β†Ύ 𝐡):𝐡⟢(𝐴 βŠ” 𝐡)
65 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inr β†Ύ 𝐡):𝐡⟢(𝐴 βŠ” 𝐡) β†’ (inr β†Ύ 𝐡) Fn 𝐡)
6664, 65mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) β†’ (inr β†Ύ 𝐡) Fn 𝐡)
67 fvco2 6938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr β†Ύ 𝐡) Fn 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡))β€˜π‘§) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
6866, 67sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡))β€˜π‘§) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
69 fvco2 6938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr β†Ύ 𝐡) Fn 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡))β€˜π‘§) = (π‘˜β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
7066, 69sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡))β€˜π‘§) = (π‘˜β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
7163, 68, 703eqtr3d 2784 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inr β†Ύ 𝐡)β€˜π‘§)) = (π‘˜β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
72 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
73 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§) β†’ (π‘˜β€˜π‘¦) = (π‘˜β€˜((inr β†Ύ 𝐡)β€˜π‘§)))
7472, 73eqeq12d 2752 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦) ↔ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜((inr β†Ύ 𝐡)β€˜π‘§)) = (π‘˜β€˜((inr β†Ύ 𝐡)β€˜π‘§))))
7571, 74syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ (𝑦 = ((inr β†Ύ 𝐡)β€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
7661, 75sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) ∧ 𝑧 ∈ 𝐡) β†’ (𝑦 = (inrβ€˜π‘§) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
7776expimpd 454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) ∧ πœ‘) β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
7877ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)) β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
7978eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘˜ ∘ (inr β†Ύ 𝐡)) = ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
8057, 79syl6bir 253 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ (πœ‘ β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
8180com23 86 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ (πœ‘ β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
82813ad2ant3 1135 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (πœ‘ β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))))
8382impcom 408 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
8483com12 32 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺 β†’ ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
85843ad2ant3 1135 . . . . . . . . . . . . . . . 16 ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))))
8685impcom 408 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
8786com12 32 . . . . . . . . . . . . . 14 ((𝑧 ∈ 𝐡 ∧ 𝑦 = (inrβ€˜π‘§)) β†’ (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
8887rexlimiva 3144 . . . . . . . . . . . . 13 (βˆƒπ‘§ ∈ 𝐡 𝑦 = (inrβ€˜π‘§) β†’ (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
8956, 88jaoi 855 . . . . . . . . . . . 12 ((βˆƒπ‘§ ∈ 𝐴 𝑦 = (inlβ€˜π‘§) ∨ βˆƒπ‘§ ∈ 𝐡 𝑦 = (inrβ€˜π‘§)) β†’ (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
90 djur 9855 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 βŠ” 𝐡) β†’ (βˆƒπ‘§ ∈ 𝐴 𝑦 = (inlβ€˜π‘§) ∨ βˆƒπ‘§ ∈ 𝐡 𝑦 = (inrβ€˜π‘§)))
9189, 90syl11 33 . . . . . . . . . . 11 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ (𝑦 ∈ (𝐴 βŠ” 𝐡) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
9291ralrimiv 3142 . . . . . . . . . 10 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ βˆ€π‘¦ ∈ (𝐴 βŠ” 𝐡)((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦))
93 ffn 6668 . . . . . . . . . . . . 13 ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) Fn (𝐴 βŠ” 𝐡))
94933ad2ant1 1133 . . . . . . . . . . . 12 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) Fn (𝐴 βŠ” 𝐡))
9594adantl 482 . . . . . . . . . . 11 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) Fn (𝐴 βŠ” 𝐡))
96 ffn 6668 . . . . . . . . . . . 12 (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 β†’ π‘˜ Fn (𝐴 βŠ” 𝐡))
97963ad2ant1 1133 . . . . . . . . . . 11 ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ π‘˜ Fn (𝐴 βŠ” 𝐡))
98 eqfnfv 6982 . . . . . . . . . . 11 (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) Fn (𝐴 βŠ” 𝐡) ∧ π‘˜ Fn (𝐴 βŠ” 𝐡)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜ ↔ βˆ€π‘¦ ∈ (𝐴 βŠ” 𝐡)((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
9995, 97, 98syl2an 596 . . . . . . . . . 10 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜ ↔ βˆ€π‘¦ ∈ (𝐴 βŠ” 𝐡)((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))β€˜π‘¦) = (π‘˜β€˜π‘¦)))
10092, 99mpbird 256 . . . . . . . . 9 (((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) ∧ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜)
101100ex 413 . . . . . . . 8 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))
102101ralrimivw 3147 . . . . . . 7 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))
10324, 102jca 512 . . . . . 6 ((πœ‘ ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺)) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜)))
104103ex 413 . . . . 5 (πœ‘ β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜))))
10521, 22, 23, 104mp3and 1464 . . . 4 (πœ‘ β†’ (((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))):(𝐴 βŠ” 𝐡)⟢𝐢 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ ((π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯)))) = π‘˜)))
1066, 17, 105rspcedvd 3583 . . 3 (πœ‘ β†’ βˆƒβ„Ž ∈ V ((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜)))
107 feq1 6649 . . . . 5 (β„Ž = π‘˜ β†’ (β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ↔ π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢))
108 coeq1 5813 . . . . . 6 (β„Ž = π‘˜ β†’ (β„Ž ∘ (inl β†Ύ 𝐴)) = (π‘˜ ∘ (inl β†Ύ 𝐴)))
109108eqeq1d 2738 . . . . 5 (β„Ž = π‘˜ β†’ ((β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ↔ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹))
110 coeq1 5813 . . . . . 6 (β„Ž = π‘˜ β†’ (β„Ž ∘ (inr β†Ύ 𝐡)) = (π‘˜ ∘ (inr β†Ύ 𝐡)))
111110eqeq1d 2738 . . . . 5 (β„Ž = π‘˜ β†’ ((β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺 ↔ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺))
112107, 109, 1113anbi123d 1436 . . . 4 (β„Ž = π‘˜ β†’ ((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ↔ (π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺)))
113112reu8 3691 . . 3 (βˆƒ!β„Ž ∈ V (β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ↔ βˆƒβ„Ž ∈ V ((β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ∧ βˆ€π‘˜ ∈ V ((π‘˜:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (π‘˜ ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (π‘˜ ∘ (inr β†Ύ 𝐡)) = 𝐺) β†’ β„Ž = π‘˜)))
114106, 113sylibr 233 . 2 (πœ‘ β†’ βˆƒ!β„Ž ∈ V (β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺))
115 reuv 3471 . 2 (βˆƒ!β„Ž ∈ V (β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺) ↔ βˆƒ!β„Ž(β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺))
116114, 115sylib 217 1 (πœ‘ β†’ βˆƒ!β„Ž(β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒ!weu 2566  βˆ€wral 3064  βˆƒwrex 3073  βˆƒ!wreu 3351  Vcvv 3445  βˆ…c0 4282  ifcif 4486   ↦ cmpt 5188   β†Ύ cres 5635   ∘ ccom 5637   Fn wfn 6491  βŸΆwf 6492  β€˜cfv 6496  1st c1st 7919  2nd c2nd 7920   βŠ” cdju 9834  inlcinl 9835  inrcinr 9836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-dju 9837  df-inl 9838  df-inr 9839
This theorem is referenced by: (None)
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