Theorem prfval 18151

Description: Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
prfval.k	⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prfval.b	⊢ 𝐵 = (Base‘𝐶)
prfval.h	⊢ 𝐻 = (Hom ‘𝐶)
prfval.c	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
prfval.d	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))

Assertion

Ref	Expression
prfval	⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)

Distinct variable groups:   𝑥,ℎ,𝑦,𝐵   𝑥,𝐶,𝑦   ℎ,𝐹,𝑥,𝑦   𝜑,ℎ,𝑥,𝑦   𝑥,𝐷,𝑦   ℎ,𝐺,𝑥,𝑦   ℎ,𝐻,𝑥,𝑦

Allowed substitution hints:   𝐶(ℎ)   𝐷(ℎ)   𝑃(𝑥,𝑦,ℎ)   𝐸(𝑥,𝑦,ℎ)

Proof of Theorem prfval

Dummy variables 𝑓 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfval.k	. 2 ⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)
2		df-prf 18127	. . . 4 ⊢ ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩))
4		fvex 6905	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
5	4	dmex 7902	. . . . 5 ⊢ dom (1st ‘𝑓) ∈ V
6	5	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) ∈ V)
7		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
8	7	fveq2d 6896	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (1st ‘𝑓) = (1st ‘𝐹))
9	8	dmeqd 5906	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = dom (1st ‘𝐹))
10		prfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
11		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		relfunc 17812	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
13		prfval.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		1st2ndbr 8028	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
15	12, 13, 14	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
16	10, 11, 15	funcf1 17816	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
17	16	fdmd 6729	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝐹) = 𝐵)
18	17	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝐹) = 𝐵)
19	9, 18	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = 𝐵)
20		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
22	21	fveq2d 6896	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑓) = (1st ‘𝐹))
23	22	fveq1d 6894	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
24		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑔 = 𝐺)
25	24	fveq2d 6896	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑔) = (1st ‘𝐺))
26	25	fveq1d 6894	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
27	23, 26	opeq12d 4882	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)
28	20, 27	mpteq12dv 5240	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩) = (𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
29		eqidd 2734	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩) = (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))
30	20, 20, 29	mpoeq123dv 7484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩)))
31	21	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑓 = 𝐹)
32	31	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘𝐹))
33	32	oveqd 7426	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
34	33	dmeqd 5906	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝑓)𝑦) = dom (𝑥(2nd ‘𝐹)𝑦))
35		prfval.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (Hom ‘𝐶)
36		eqid 2733	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
37	15	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
38		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
39		simpr 486	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
40	10, 35, 36, 37, 38, 39	funcf2 17818	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝐹)𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
41	40	fdmd 6729	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝐹)𝑦) = (𝑥𝐻𝑦))
42	34, 41	eqtrd 2773	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝑓)𝑦) = (𝑥𝐻𝑦))
43	33	fveq1d 6894	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(2nd ‘𝑓)𝑦)‘ℎ) = ((𝑥(2nd ‘𝐹)𝑦)‘ℎ))
44	24	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑔 = 𝐺)
45	44	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑔) = (2nd ‘𝐺))
46	45	oveqd 7426	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑔)𝑦) = (𝑥(2nd ‘𝐺)𝑦))
47	46	fveq1d 6894	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(2nd ‘𝑔)𝑦)‘ℎ) = ((𝑥(2nd ‘𝐺)𝑦)‘ℎ))
48	43, 47	opeq12d 4882	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩ = ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)
49	42, 48	mpteq12dv 5240	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩) = (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))
50	49	3impa 1111	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩) = (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))
51	50	mpoeq3dva 7486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)))
52	30, 51	eqtrd 2773	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)))
53	28, 52	opeq12d 4882	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
54	6, 19, 53	csbied2 3934	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
55	13	elexd 3495	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
56		prfval.d	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
57	56	elexd 3495	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
58		opex 5465	. . . 4 ⊢ ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ ∈ V
59	58	a1i 11	. . 3 ⊢ (𝜑 → ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ ∈ V)
60	3, 54, 55, 57, 59	ovmpod 7560	. 2 ⊢ (𝜑 → (𝐹 ⟨,⟩F 𝐺) = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
61	1, 60	eqtrid 2785	1 ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⦋csb 3894  ⟨cop 4635   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  Rel wrel 5682  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974  Basecbs 17144  Hom chom 17208   Func cfunc 17804   ⟨,⟩_F cprf 18123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-ixp 8892  df-func 17808  df-prf 18127

This theorem is referenced by:  prf1  18152  prf2fval  18153  prfcl  18155  prf1st  18156  prf2nd  18157  1st2ndprf  18158