Theorem prfval 18102

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 18078	. . . 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 6835	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
5	4	dmex 7839	. . . . 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 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (1st ‘𝑓) = (1st ‘𝐹))
9	8	dmeqd 5845	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = dom (1st ‘𝐹))
10		prfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
11		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		relfunc 17766	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
13		prfval.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		1st2ndbr 7974	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
15	12, 13, 14	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
16	10, 11, 15	funcf1 17770	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
17	16	fdmd 6661	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝐹) = 𝐵)
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝐹) = 𝐵)
19	9, 18	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = 𝐵)
20		simpr 484	. . . . . 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 6826	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑓) = (1st ‘𝐹))
23	22	fveq1d 6824	. . . . . . 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 6826	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑔) = (1st ‘𝐺))
26	25	fveq1d 6824	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
27	23, 26	opeq12d 4833	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)
28	20, 27	mpteq12dv 5178	. . . . 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 2732	. . . . . . 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 7421	. . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑓 = 𝐹)
32	31	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘𝐹))
33	32	oveqd 7363	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
34	33	dmeqd 5845	. . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
37	15	ad4antr 732	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
40	10, 35, 36, 37, 38, 39	funcf2 17772	. . . . . . . . . . 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 6661	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝐹)𝑦) = (𝑥𝐻𝑦))
42	34, 41	eqtrd 2766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝑓)𝑦) = (𝑥𝐻𝑦))
43	33	fveq1d 6824	. . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑔 = 𝐺)
45	44	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑔) = (2nd ‘𝐺))
46	45	oveqd 7363	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑔)𝑦) = (𝑥(2nd ‘𝐺)𝑦))
47	46	fveq1d 6824	. . . . . . . . . 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 4833	. . . . . . . . 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 5178	. . . . . . . 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 1109	. . . . . . 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 7423	. . . . . 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 2766	. . . . 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 4833	. . . 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 3887	. . 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 3460	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
56		prfval.d	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
57	56	elexd 3460	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
58		opex 5404	. . . 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 7498	. 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 2778	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 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3850  ⟨cop 4582   class class class wbr 5091   ↦ cmpt 5172  dom cdm 5616  Rel wrel 5621  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  Hom chom 17169   Func cfunc 17758   ⟨,⟩_F cprf 18074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-ixp 8822  df-func 17762  df-prf 18078

This theorem is referenced by:  prf1  18103  prf2fval  18104  prfcl  18106  prf1st  18107  prf2nd  18108  1st2ndprf  18109