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Theorem updjud 9352
Description: Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9320 and df-inr 9321, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9321 (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 (𝜑𝐹:𝐴𝐶)
updjud.g (𝜑𝐺:𝐵𝐶)
updjud.a (𝜑𝐴𝑉)
updjud.b (𝜑𝐵𝑊)
Assertion
Ref Expression
updjud (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
Distinct variable groups:   𝐴,   𝐵,   𝐶,   ,𝐹   ,𝐺   𝜑,
Allowed substitution hints:   𝑉()   𝑊()

Proof of Theorem updjud
Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 updjud.a . . . . . 6 (𝜑𝐴𝑉)
2 updjud.b . . . . . 6 (𝜑𝐵𝑊)
31, 2jca 512 . . . . 5 (𝜑 → (𝐴𝑉𝐵𝑊))
4 djuex 9326 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
5 mptexg 6979 . . . . 5 ((𝐴𝐵) ∈ V → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∈ V)
63, 4, 53syl 18 . . . 4 (𝜑 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∈ V)
7 feq1 6492 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (:(𝐴𝐵)⟶𝐶 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶))
8 coeq1 5727 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)))
98eqeq1d 2828 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (( ∘ (inl ↾ 𝐴)) = 𝐹 ↔ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹))
10 coeq1 5727 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)))
1110eqeq1d 2828 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (( ∘ (inr ↾ 𝐵)) = 𝐺 ↔ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
127, 9, 113anbi123d 1429 . . . . . 6 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)))
13 eqeq1 2830 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( = 𝑘 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))
1413imbi2d 342 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘) ↔ ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘)))
1514ralbidv 3202 . . . . . 6 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘) ↔ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘)))
1612, 15anbi12d 630 . . . . 5 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘)) ↔ (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))))
1716adantl 482 . . . 4 ((𝜑 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))) → (((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘)) ↔ (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))))
18 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
19 updjud.g . . . . . 6 (𝜑𝐺:𝐵𝐶)
20 eqid 2826 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
2118, 19, 20updjudhf 9349 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶)
2218, 19, 20updjudhcoinlf 9350 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹)
2318, 19, 20updjudhcoinrg 9351 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)
24 simpr 485 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
25 eqeq2 2838 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
26 fvres 6686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐴 → ((inl ↾ 𝐴)‘𝑧) = (inl‘𝑧))
2726eqcomd 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐴 → (inl‘𝑧) = ((inl ↾ 𝐴)‘𝑧))
2827eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐴 → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
2928adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
30 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
3130ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
32 inlresf 9332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
33 ffn 6511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
3432, 33mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → (inl ↾ 𝐴) Fn 𝐴)
35 fvco2 6755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl ↾ 𝐴) Fn 𝐴𝑧𝐴) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
3634, 35sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
37 fvco2 6755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl ↾ 𝐴) Fn 𝐴𝑧𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
3834, 37sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
3931, 36, 383eqtr3d 2869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
40 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
41 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (𝑘𝑦) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
4240, 41eqeq12d 2842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦) ↔ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧))))
4339, 42syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
4429, 43sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧𝐴) → (𝑦 = (inl‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
4544expimpd 454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
4645ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
4746eqcoms 2834 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
4825, 47syl6bir 255 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
4948com23 86 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
50493ad2ant2 1128 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
5150impcom 408 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
5251com12 32 . . . . . . . . . . . . . . . . 17 ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
53523ad2ant2 1128 . . . . . . . . . . . . . . . 16 ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
5453impcom 408 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐴𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
5554com12 32 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦 = (inl‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
5655rexlimiva 3286 . . . . . . . . . . . . 13 (∃𝑧𝐴 𝑦 = (inl‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
57 eqeq2 2838 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
58 fvres 6686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐵 → ((inr ↾ 𝐵)‘𝑧) = (inr‘𝑧))
5958eqcomd 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐵 → (inr‘𝑧) = ((inr ↾ 𝐵)‘𝑧))
6059eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐵 → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
6160adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
62 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
6362ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
64 inrresf 9334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
65 ffn 6511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) → (inr ↾ 𝐵) Fn 𝐵)
6664, 65mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → (inr ↾ 𝐵) Fn 𝐵)
67 fvco2 6755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr ↾ 𝐵) Fn 𝐵𝑧𝐵) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
6866, 67sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
69 fvco2 6755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr ↾ 𝐵) Fn 𝐵𝑧𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7066, 69sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7163, 68, 703eqtr3d 2869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
72 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
73 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (𝑘𝑦) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7472, 73eqeq12d 2842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦) ↔ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧))))
7571, 74syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
7661, 75sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧𝐵) → (𝑦 = (inr‘𝑧) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
7776expimpd 454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
7877ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
7978eqcoms 2834 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
8057, 79syl6bir 255 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
8180com23 86 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
82813ad2ant3 1129 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))))
8382impcom 408 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
8483com12 32 . . . . . . . . . . . . . . . . 17 ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
85843ad2ant3 1129 . . . . . . . . . . . . . . . 16 ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))))
8685impcom 408 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧𝐵𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
8786com12 32 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
8887rexlimiva 3286 . . . . . . . . . . . . 13 (∃𝑧𝐵 𝑦 = (inr‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
8956, 88jaoi 853 . . . . . . . . . . . 12 ((∃𝑧𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧𝐵 𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
90 djur 9337 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴𝐵) → (∃𝑧𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧𝐵 𝑦 = (inr‘𝑧)))
9189, 90syl11 33 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑦 ∈ (𝐴𝐵) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
9291ralrimiv 3186 . . . . . . . . . 10 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑦 ∈ (𝐴𝐵)((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦))
93 ffn 6511 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵))
94933ad2ant1 1127 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵))
9594adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵))
96 ffn 6511 . . . . . . . . . . . 12 (𝑘:(𝐴𝐵)⟶𝐶𝑘 Fn (𝐴𝐵))
97963ad2ant1 1127 . . . . . . . . . . 11 ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → 𝑘 Fn (𝐴𝐵))
98 eqfnfv 6798 . . . . . . . . . . 11 (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵) ∧ 𝑘 Fn (𝐴𝐵)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴𝐵)((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
9995, 97, 98syl2an 595 . . . . . . . . . 10 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴𝐵)((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))‘𝑦) = (𝑘𝑦)))
10092, 99mpbird 258 . . . . . . . . 9 (((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘)
101100ex 413 . . . . . . . 8 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))
102101ralrimivw 3188 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))
10324, 102jca 512 . . . . . 6 ((𝜑 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘)))
104103ex 413 . . . . 5 (𝜑 → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))))
10521, 22, 23, 104mp3and 1457 . . . 4 (𝜑 → (((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘)))
1066, 17, 105rspcedvd 3630 . . 3 (𝜑 → ∃ ∈ V ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘)))
107 feq1 6492 . . . . 5 ( = 𝑘 → (:(𝐴𝐵)⟶𝐶𝑘:(𝐴𝐵)⟶𝐶))
108 coeq1 5727 . . . . . 6 ( = 𝑘 → ( ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)))
109108eqeq1d 2828 . . . . 5 ( = 𝑘 → (( ∘ (inl ↾ 𝐴)) = 𝐹 ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
110 coeq1 5727 . . . . . 6 ( = 𝑘 → ( ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)))
111110eqeq1d 2828 . . . . 5 ( = 𝑘 → (( ∘ (inr ↾ 𝐵)) = 𝐺 ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
112107, 109, 1113anbi123d 1429 . . . 4 ( = 𝑘 → ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)))
113112reu8 3728 . . 3 (∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃ ∈ V ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → = 𝑘)))
114106, 113sylibr 235 . 2 (𝜑 → ∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
115 reuv 3527 . 2 (∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
116114, 115sylib 219 1 (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  ∃!weu 2651  wral 3143  wrex 3144  ∃!wreu 3145  Vcvv 3500  c0 4295  ifcif 4470  cmpt 5143  cres 5556  ccom 5558   Fn wfn 6347  wf 6348  cfv 6352  1st c1st 7678  2nd c2nd 7679  cdju 9316  inlcinl 9317  inrcinr 9318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1st 7680  df-2nd 7681  df-1o 8093  df-dju 9319  df-inl 9320  df-inr 9321
This theorem is referenced by: (None)
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