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Theorem updjud 7081

Description: Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)

**Hypotheses**
Ref	Expression
updjud.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
updjud.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
updjud.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
updjud.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)

**Assertion**
Ref	Expression
updjud	⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))

Distinct variable groups: 𝐴,ℎ 𝐵,ℎ 𝐶,ℎ ℎ,𝐹 ℎ,𝐺 𝜑,ℎ Allowed substitution hints: 𝑉(ℎ) 𝑊(ℎ)

Proof of Theorem updjud Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		updjud.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		updjud.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3	1, 2	jca 306	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
4		djuex 7042	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
5		mptexg 5742	. . . . 5 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∈ V)
6	3, 4, 5	3syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∈ V)
7		feq1 5349	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ↔ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶))
8		coeq1 4785	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (ℎ ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)))
9	8	eqeq1d 2186	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → ((ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹))
10		coeq1 4785	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (ℎ ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)))
11	10	eqeq1d 2186	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → ((ℎ ∘ (inr ↾ 𝐵)) = 𝐺 ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
12	7, 9, 11	3anbi123d 1312	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → ((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)))
13		eqeq1 2184	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (ℎ = 𝑘 ↔ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))
14	13	imbi2d 230	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘) ↔ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘)))
15	14	ralbidv 2477	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘) ↔ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘)))
16	12, 15	anbi12d 473	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) → (((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘)) ↔ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))))
17	16	adantl 277	. . . 4 ⊢ ((𝜑 ∧ ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))) → (((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘)) ↔ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))))
18		updjud.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
19		updjud.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
20		eqid 2177	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))
21	18, 19, 20	updjudhf 7078	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶)
22	18, 19, 20	updjudhcoinlf 7079	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹)
23	18, 19, 20	updjudhcoinrg 7080	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)
24		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
25		eqeq2 2187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
26		fvres 5540	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑧) = (inl‘𝑧))
27	26	eqcomd 2183	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝐴 → (inl‘𝑧) = ((inl ↾ 𝐴)‘𝑧))
28	27	eqeq2d 2189	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
29	28	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
30		fveq1 5515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
31	30	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
32		inlresf1 7060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)
33		f1fn 5424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
34	32, 33	mp1i 10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → (inl ↾ 𝐴) Fn 𝐴)
35		fvco2 5586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
36	34, 35	sylan 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
37		fvco2 5586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
38	34, 37	sylan 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
39	31, 36, 38	3eqtr3d 2218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
40		fveq2 5516	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
41		fveq2 5516	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (𝑘‘𝑦) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
42	40, 41	eqeq12d 2192	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧))))
43	39, 42	syl5ibrcom 157	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
44	29, 43	sylbid 150	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (inl‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
45	44	expimpd 363	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
46	45	ex 115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
47	46	eqcoms 2180	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
48	25, 47	syl6bir 164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
49	48	com23 78	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
50	49	3ad2ant2 1019	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
51	50	impcom 125	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
52	51	com12 30	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
53	52	3ad2ant2 1019	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
54	53	impcom 125	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
55	54	com12 30	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
56	55	rexlimiva 2589	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
57		eqeq2 2187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
58		fvres 5540	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑧) = (inr‘𝑧))
59	58	eqcomd 2183	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝐵 → (inr‘𝑧) = ((inr ↾ 𝐵)‘𝑧))
60	59	eqeq2d 2189	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
61	60	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
62		fveq1 5515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
63	62	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
64		inrresf1 7061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵)
65		f1fn 5424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
66	64, 65	mp1i 10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → (inr ↾ 𝐵) Fn 𝐵)
67		fvco2 5586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
68	66, 67	sylan 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
69		fvco2 5586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
70	66, 69	sylan 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
71	63, 68, 70	3eqtr3d 2218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
72		fveq2 5516	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
73		fveq2 5516	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (𝑘‘𝑦) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
74	72, 73	eqeq12d 2192	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧))))
75	71, 74	syl5ibrcom 157	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
76	61, 75	sylbid 150	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = (inr‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
77	76	expimpd 363	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
78	77	ex 115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
79	78	eqcoms 2180	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
80	57, 79	syl6bir 164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
81	80	com23 78	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
82	81	3ad2ant3 1020	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
83	82	impcom 125	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
84	83	com12 30	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
85	84	3ad2ant3 1020	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
86	85	impcom 125	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
87	86	com12 30	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
88	87	rexlimiva 2589	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
89	56, 88	jaoi 716	. . . . . . . . . . . 12 ⊢ ((∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
90		djur 7068	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧)))
91	90	biimpi 120	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴 ⊔ 𝐵) → (∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧)))
92	89, 91	syl11 31	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑦 ∈ (𝐴 ⊔ 𝐵) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
93	92	ralrimiv 2549	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))
94		ffn 5366	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
95	94	3ad2ant1 1018	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
96	95	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
97		ffn 5366	. . . . . . . . . . . 12 ⊢ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 → 𝑘 Fn (𝐴 ⊔ 𝐵))
98	97	3ad2ant1 1018	. . . . . . . . . . 11 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → 𝑘 Fn (𝐴 ⊔ 𝐵))
99		eqfnfv 5614	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵) ∧ 𝑘 Fn (𝐴 ⊔ 𝐵)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
100	96, 98, 99	syl2an 289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
101	93, 100	mpbird 167	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘)
102	101	ex 115	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))
103	102	ralrimivw 2551	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))
104	24, 103	jca 306	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘)))
105	104	ex 115	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘))))
106	21, 22, 23, 105	mp3and 1340	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥)))) = 𝑘)))
107	6, 17, 106	rspcedvd 2848	. . 3 ⊢ (𝜑 → ∃ℎ ∈ V ((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘)))
108		feq1 5349	. . . . 5 ⊢ (ℎ = 𝑘 → (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ↔ 𝑘:(𝐴 ⊔ 𝐵)⟶𝐶))
109		coeq1 4785	. . . . . 6 ⊢ (ℎ = 𝑘 → (ℎ ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)))
110	109	eqeq1d 2186	. . . . 5 ⊢ (ℎ = 𝑘 → ((ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
111		coeq1 4785	. . . . . 6 ⊢ (ℎ = 𝑘 → (ℎ ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)))
112	111	eqeq1d 2186	. . . . 5 ⊢ (ℎ = 𝑘 → ((ℎ ∘ (inr ↾ 𝐵)) = 𝐺 ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
113	108, 110, 112	3anbi123d 1312	. . . 4 ⊢ (ℎ = 𝑘 → ((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ↔ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)))
114	113	reu8 2934	. . 3 ⊢ (∃!ℎ ∈ V (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃ℎ ∈ V ((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ℎ = 𝑘)))
115	107, 114	sylibr 134	. 2 ⊢ (𝜑 → ∃!ℎ ∈ V (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
116		reuv 2757	. 2 ⊢ (∃!ℎ ∈ V (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
117	115, 116	sylib 122	1 ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 ∧ w3a 978 = wceq 1353 ∃!weu 2026 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∃!wreu 2457 Vcvv 2738 ∅c0 3423 ifcif 3535 ↦ cmpt 4065 ↾ cres 4629 ∘ ccom 4631 Fn wfn 5212 ⟶wf 5213 –1-1→wf1 5214 ‘cfv 5217 1^st c1st 6139 2^nd c2nd 6140 ⊔ cdju 7036 inlcinl 7044 inrcinr 7045

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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-1o 6417 df-dju 7037 df-inl 7046 df-inr 7047

This theorem is referenced by: (None)

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