Theorem updjud 10003

Description: Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9971 and df-inr 9972, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9972 (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	⊢ (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 511	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
4		djuex 9977	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
5		mptexg 7258	. . . . 5 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∈ V)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∈ V)
7		feq1 6728	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ↔ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶))
8		coeq1 5882	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → (ℎ ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)))
9	8	eqeq1d 2742	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → ((ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹))
10		coeq1 5882	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → (ℎ ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)))
11	10	eqeq1d 2742	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → ((ℎ ∘ (inr ↾ 𝐵)) = 𝐺 ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
12	7, 9, 11	3anbi123d 1436	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → ((ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)))
13		eqeq1 2744	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) → (ℎ = 𝑘 ↔ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))
14	13	imbi2d 340	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘)))
15	14	ralbidv 3184	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘)))
16	12, 15	anbi12d 631	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))))
17	16	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ℎ = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))))
18		updjud.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
19		updjud.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
20		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))
21	18, 19, 20	updjudhf 10000	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶)
22	18, 19, 20	updjudhcoinlf 10001	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹)
23	18, 19, 20	updjudhcoinrg 10002	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)
24		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺))
25		eqeq2 2752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
26		fvres 6939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑧) = (inl‘𝑧))
27	26	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝐴 → (inl‘𝑧) = ((inl ↾ 𝐴)‘𝑧))
28	27	eqeq2d 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
29	28	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (inl‘𝑧) ↔ 𝑦 = ((inl ↾ 𝐴)‘𝑧)))
30		fveq1 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
31	30	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧))
32		inlresf 9983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
33		ffn 6747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
34	32, 33	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → (inl ↾ 𝐴) Fn 𝐴)
35		fvco2 7019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
36	34, 35	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
37		fvco2 7019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
38	34, 37	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → ((𝑘 ∘ (inl ↾ 𝐴))‘𝑧) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
39	31, 36, 38	3eqtr3d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
40		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)))
41		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (𝑘‘𝑦) = (𝑘‘((inl ↾ 𝐴)‘𝑧)))
42	40, 41	eqeq12d 2756	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ((inl ↾ 𝐴)‘𝑧) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inl ↾ 𝐴)‘𝑧)) = (𝑘‘((inl ↾ 𝐴)‘𝑧))))
43	39, 42	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((inl ↾ 𝐴)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
44	29, 43	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (inl‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
45	44	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) ∧ 𝜑) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
46	45	ex 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
47	46	eqcoms 2748	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∘ (inl ↾ 𝐴)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
48	25, 47	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
49	48	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
50	49	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
51	50	impcom 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
52	51	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
53	52	3ad2ant2 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
54	53	impcom 407	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
55	54	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
56	55	rexlimiva 3153	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
57		eqeq2 2752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
58		fvres 6939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑧) = (inr‘𝑧))
59	58	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝐵 → (inr‘𝑧) = ((inr ↾ 𝐵)‘𝑧))
60	59	eqeq2d 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = (inr‘𝑧) ↔ 𝑦 = ((inr ↾ 𝐵)‘𝑧)))
62		fveq1 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
63	62	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧))
64		inrresf 9985	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
65		ffn 6747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
66	64, 65	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → (inr ↾ 𝐵) Fn 𝐵)
67		fvco2 7019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
68	66, 67	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵))‘𝑧) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
69		fvco2 7019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
70	66, 69	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → ((𝑘 ∘ (inr ↾ 𝐵))‘𝑧) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
71	63, 68, 70	3eqtr3d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
72		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)))
73		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (𝑘‘𝑦) = (𝑘‘((inr ↾ 𝐵)‘𝑧)))
74	72, 73	eqeq12d 2756	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ((inr ↾ 𝐵)‘𝑧) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦) ↔ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘((inr ↾ 𝐵)‘𝑧)) = (𝑘‘((inr ↾ 𝐵)‘𝑧))))
75	71, 74	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = ((inr ↾ 𝐵)‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
76	61, 75	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) ∧ 𝑧 ∈ 𝐵) → (𝑦 = (inr‘𝑧) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
77	76	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) ∧ 𝜑) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
78	77	ex 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
79	78	eqcoms 2748	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∘ (inr ↾ 𝐵)) = ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
80	57, 79	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
81	80	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺 → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
82	81	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝜑 → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))))
83	82	impcom 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
84	83	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∘ (inr ↾ 𝐵)) = 𝐺 → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
85	84	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))))
86	85	impcom 407	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
87	86	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
88	87	rexlimiva 3153	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
89	56, 88	jaoi 856	. . . . . . . . . . . 12 ⊢ ((∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧)) → (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
90		djur 9988	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴 ⊔ 𝐵) → (∃𝑧 ∈ 𝐴 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 𝑦 = (inr‘𝑧)))
91	89, 90	syl11 33	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑦 ∈ (𝐴 ⊔ 𝐵) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
92	91	ralrimiv 3151	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦))
93		ffn 6747	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
94	93	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
95	94	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵))
96		ffn 6747	. . . . . . . . . . . 12 ⊢ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 → 𝑘 Fn (𝐴 ⊔ 𝐵))
97	96	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → 𝑘 Fn (𝐴 ⊔ 𝐵))
98		eqfnfv 7064	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) Fn (𝐴 ⊔ 𝐵) ∧ 𝑘 Fn (𝐴 ⊔ 𝐵)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
99	95, 97, 98	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘 ↔ ∀𝑦 ∈ (𝐴 ⊔ 𝐵)((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))‘𝑦) = (𝑘‘𝑦)))
100	92, 99	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) ∧ (𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺)) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘)
101	100	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))
102	101	ralrimivw 3156	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))
103	24, 102	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺)) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘)))
104	103	ex 412	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘))))
105	21, 22, 23, 104	mp3and 1464	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))):(𝐴 ⊔ 𝐵)⟶𝐶 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ∘ (inr ↾ 𝐵)) = 𝐺) ∧ ∀𝑘 ∈ V ((𝑘:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺) → (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) = 𝑘)))
106	6, 17, 105	rspcedvd 3637	. . 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)))
107		feq1 6728	. . . . 5 ⊢ (ℎ = 𝑘 → (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ↔ 𝑘:(𝐴 ⊔ 𝐵)⟶𝐶))
108		coeq1 5882	. . . . . 6 ⊢ (ℎ = 𝑘 → (ℎ ∘ (inl ↾ 𝐴)) = (𝑘 ∘ (inl ↾ 𝐴)))
109	108	eqeq1d 2742	. . . . 5 ⊢ (ℎ = 𝑘 → ((ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ↔ (𝑘 ∘ (inl ↾ 𝐴)) = 𝐹))
110		coeq1 5882	. . . . . 6 ⊢ (ℎ = 𝑘 → (ℎ ∘ (inr ↾ 𝐵)) = (𝑘 ∘ (inr ↾ 𝐵)))
111	110	eqeq1d 2742	. . . . 5 ⊢ (ℎ = 𝑘 → ((ℎ ∘ (inr ↾ 𝐵)) = 𝐺 ↔ (𝑘 ∘ (inr ↾ 𝐵)) = 𝐺))
112	107, 109, 111	3anbi123d 1436	. . . 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)))
113	112	reu8 3755	. . 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)))
114	106, 113	sylibr 234	. 2 ⊢ (𝜑 → ∃!ℎ ∈ V (ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
115		reuv 3518	. 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))
116	114, 115	sylib 218	1 ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  Vcvv 3488  ∅c0 4352  ifcif 4548   ↦ cmpt 5249   ↾ cres 5702   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029   ⊔ cdju 9967  inlcinl 9968  inrcinr 9969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-dju 9970  df-inl 9971  df-inr 9972

This theorem is referenced by: (None)