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Theorem updjud 7071
Description: Universal property of the disjoint union. (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 306 . . . . 5 (𝜑 → (𝐴𝑉𝐵𝑊))
4 djuex 7032 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
5 mptexg 5733 . . . . 5 ((𝐴𝐵) ∈ V → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∈ V)
63, 4, 53syl 17 . . . 4 (𝜑 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∈ V)
7 feq1 5340 . . . . . . 7 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → (:(𝐴𝐵)⟶𝐶 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶))
8 coeq1 4777 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)))
98eqeq1d 2184 . . . . . . 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 4777 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)))
1110eqeq1d 2184 . . . . . . 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 1312 . . . . . 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 2182 . . . . . . . 8 ( = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) → ( = 𝑘 ↔ (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = 𝑘))
1413imbi2d 230 . . . . . . 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 2475 . . . . . 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 473 . . . . 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 277 . . . 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 2175 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
2118, 19, 20updjudhf 7068 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶)
2218, 19, 20updjudhcoinlf 7069 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹)
2318, 19, 20updjudhcoinrg 7070 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)
24 simpr 110 . . . . . . 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 2185 . . . . . . . . . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐴 → ((inl ↾ 𝐴)‘𝑧) = (inl‘𝑧))
2726eqcomd 2181 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐴 → (inl‘𝑧) = ((inl ↾ 𝐴)‘𝑧))
2827eqeq2d 2187 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐴 → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
2928adantl 277 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 inlresf1 7050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
33 f1fn 5415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inl ↾ 𝐴):𝐴1-1→(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
3432, 33mp1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → (inl ↾ 𝐴) Fn 𝐴)
35 fvco2 5577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inl ↾ 𝐴) Fn 𝐴𝑧𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
3834, 37sylan 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2216 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5507 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5507 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (𝑘𝑦) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
4240, 41eqeq12d 2190 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 157 . . . . . . . . . . . . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . . . . . . . . . . . . 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 363 . . . . . . . . . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . . . . . . . . . . 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 2178 . . . . . . . . . . . . . . . . . . . . . 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 164 . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . . . . . . . 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 125 . . . . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . . . . 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 125 . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . 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 2587 . . . . . . . . . . . . 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 2185 . . . . . . . . . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧𝐵 → ((inr ↾ 𝐵)‘𝑧) = (inr‘𝑧))
5958eqcomd 2181 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧𝐵 → (inr‘𝑧) = ((inr ↾ 𝐵)‘𝑧))
6059eqeq2d 2187 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝐵 → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
6160adantl 277 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 inrresf1 7051 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)
65 f1fn 5415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((inr ↾ 𝐵):𝐵1-1→(𝐴𝐵) → (inr ↾ 𝐵) Fn 𝐵)
6664, 65mp1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → (inr ↾ 𝐵) Fn 𝐵)
67 fvco2 5577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((inr ↾ 𝐵) Fn 𝐵𝑧𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7066, 69sylan 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2216 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5507 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5507 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (𝑘𝑦) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
7472, 73eqeq12d 2190 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 157 . . . . . . . . . . . . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . . . . . . . . . . . . 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 363 . . . . . . . . . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . . . . . . . . . . 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 2178 . . . . . . . . . . . . . . . . . . . . . 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 164 . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . . . . 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 125 . . . . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . 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 125 . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . 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 2587 . . . . . . . . . . . . 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 716 . . . . . . . . . . . 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 7058 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴𝐵) ↔ (∃𝑧𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧𝐵 𝑦 = (inr‘𝑧)))
9190biimpi 120 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴𝐵) → (∃𝑧𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧𝐵 𝑦 = (inr‘𝑧)))
9289, 91syl11 31 . . . . . . . . . . 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)))
9392ralrimiv 2547 . . . . . . . . . 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))
94 ffn 5357 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))):(𝐴𝐵)⟶𝐶 → (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))) Fn (𝐴𝐵))
95943ad2ant1 1018 . . . . . . . . . . . 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))
9695adantl 277 . . . . . . . . . . 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))
97 ffn 5357 . . . . . . . . . . . 12 (𝑘:(𝐴𝐵)⟶𝐶𝑘 Fn (𝐴𝐵))
98973ad2ant1 1018 . . . . . . . . . . 11 ((𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → 𝑘 Fn (𝐴𝐵))
99 eqfnfv 5605 . . . . . . . . . . 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)))
10096, 98, 99syl2an 289 . . . . . . . . . 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)))
10193, 100mpbird 167 . . . . . . . . 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)
102101ex 115 . . . . . . . 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))
103102ralrimivw 2549 . . . . . . 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))
10424, 103jca 306 . . . . . 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)))
105104ex 115 . . . . 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))))
10621, 22, 23, 105mp3and 1340 . . . 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)))
1076, 17, 106rspcedvd 2845 . . 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)))
108 feq1 5340 . . . . 5 ( = 𝑘 → (:(𝐴𝐵)⟶𝐶𝑘:(𝐴𝐵)⟶𝐶))
109 coeq1 4777 . . . . . 6 ( = 𝑘 → ( ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)))
110109eqeq1d 2184 . . . . 5 ( = 𝑘 → (( ∘ (inl ↾ 𝐴)) = 𝐹 ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
111 coeq1 4777 . . . . . 6 ( = 𝑘 → ( ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)))
112111eqeq1d 2184 . . . . 5 ( = 𝑘 → (( ∘ (inr ↾ 𝐵)) = 𝐺 ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
113108, 110, 1123anbi123d 1312 . . . 4 ( = 𝑘 → ((:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ (𝑘:(𝐴𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)))
114113reu8 2931 . . 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)))
115107, 114sylibr 134 . 2 (𝜑 → ∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
116 reuv 2754 . 2 (∃! ∈ V (:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
117115, 116sylib 122 1 (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  ∃!weu 2024  wcel 2146  wral 2453  wrex 2454  ∃!wreu 2455  Vcvv 2735  c0 3420  ifcif 3532  cmpt 4059  cres 4622  ccom 4624   Fn wfn 5203  wf 5204  1-1wf1 5205  cfv 5208  1st c1st 6129  2nd c2nd 6130  cdju 7026  inlcinl 7034  inrcinr 7035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037
This theorem is referenced by: (None)
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