Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  updjud Structured version   Visualization version   GIF version

Theorem updjud 9401
 Description: Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9369 and df-inr 9370, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9370 (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 515 . . . . 5 (𝜑 → (𝐴𝑉𝐵𝑊))
4 djuex 9375 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
5 mptexg 6980 . . . . 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 6483 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (:(𝐴𝐵)⟶𝐶 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶))
8 coeq1 5702 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)))
98eqeq1d 2760 . . . . . . 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 5702 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)))
1110eqeq1d 2760 . . . . . . 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 1433 . . . . . 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 2762 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( = 𝑘 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))
1413imbi2d 344 . . . . . . 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 3126 . . . . . 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 633 . . . . 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 485 . . . 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 2758 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
2118, 19, 20updjudhf 9398 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶)
2218, 19, 20updjudhcoinlf 9399 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹)
2318, 19, 20updjudhcoinrg 9400 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)
24 simpr 488 . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . . . . 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 6681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐴 → ((inl ↾ 𝐴)‘𝑧) = (inl‘𝑧))
2726eqcomd 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐴 → (inl‘𝑧) = ((inl ↾ 𝐴)‘𝑧))
2827eqeq2d 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐴 → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
2928adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9381 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
33 ffn 6502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl ↾ 𝐴) Fn 𝐴𝑧𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
3834, 37sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (𝑘𝑦) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
4240, 41eqeq12d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 250 . . . . . . . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 3205 . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . . . . 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 6681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐵 → ((inr ↾ 𝐵)‘𝑧) = (inr‘𝑧))
5958eqcomd 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐵 → (inr‘𝑧) = ((inr ↾ 𝐵)‘𝑧))
6059eqeq2d 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐵 → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
6160adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
65 ffn 6502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr ↾ 𝐵) Fn 𝐵𝑧𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7066, 69sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (𝑘𝑦) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7472, 73eqeq12d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 250 . . . . . . . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 3205 . . . . . . . . . . . . 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 854 . . . . . . . . . . . 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 9386 . . . . . . . . . . . 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 3112 . . . . . . . . . 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 6502 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵))
94933ad2ant1 1130 . . . . . . . . . . . 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 485 . . . . . . . . . . 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 6502 . . . . . . . . . . . 12 (𝑘:(𝐴𝐵)⟶𝐶𝑘 Fn (𝐴𝐵))
97963ad2ant1 1130 . . . . . . . . . . 11 ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → 𝑘 Fn (𝐴𝐵))
98 eqfnfv 6797 . . . . . . . . . . 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 598 . . . . . . . . . 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 260 . . . . . . . . 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 416 . . . . . . . 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 3114 . . . . . . 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 515 . . . . . 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 416 . . . . 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 1461 . . . 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 3546 . . 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 6483 . . . . 5 ( = 𝑘 → (:(𝐴𝐵)⟶𝐶𝑘:(𝐴𝐵)⟶𝐶))
108 coeq1 5702 . . . . . 6 ( = 𝑘 → ( ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)))
109108eqeq1d 2760 . . . . 5 ( = 𝑘 → (( ∘ (inl ↾ 𝐴)) = 𝐹 ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
110 coeq1 5702 . . . . . 6 ( = 𝑘 → ( ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)))
111110eqeq1d 2760 . . . . 5 ( = 𝑘 → (( ∘ (inr ↾ 𝐵)) = 𝐺 ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
112107, 109, 1113anbi123d 1433 . . . 4 ( = 𝑘 → ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)))
113112reu8 3649 . . 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 237 . 2 (𝜑 → ∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
115 reuv 3437 . 2 (∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
116114, 115sylib 221 1 (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃!weu 2587  ∀wral 3070  ∃wrex 3071  ∃!wreu 3072  Vcvv 3409  ∅c0 4227  ifcif 4423   ↦ cmpt 5115   ↾ cres 5529   ∘ ccom 5531   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  1st c1st 7696  2nd c2nd 7697   ⊔ cdju 9365  inlcinl 9366  inrcinr 9367 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7585  df-1st 7698  df-2nd 7699  df-1o 8117  df-dju 9368  df-inl 9369  df-inr 9370 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator