Theorem prfval 18256

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 18232	. . . 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 6896	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
5	4	dmex 7907	. . . . 5 ⊢ dom (1st ‘𝑓) ∈ V
6	5	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) ∈ V)
7		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
8	7	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (1st ‘𝑓) = (1st ‘𝐹))
9	8	dmeqd 5897	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = dom (1st ‘𝐹))
10		prfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
11		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		relfunc 17920	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
13		prfval.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		1st2ndbr 8040	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
15	12, 13, 14	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
16	10, 11, 15	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
17	16	fdmd 6718	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝐹) = 𝐵)
18	17	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝐹) = 𝐵)
19	9, 18	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → dom (1st ‘𝑓) = 𝐵)
20		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21		simplrl 788	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
22	21	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑓) = (1st ‘𝐹))
23	22	fveq1d 6885	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
24		simplrr 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑔 = 𝐺)
25	24	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st ‘𝑔) = (1st ‘𝐺))
26	25	fveq1d 6885	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
27	23, 26	opeq12d 4847	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)
28	20, 27	mpteq12dv 5199	. . . . 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 2764	. . . . . . 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 7487	. . . . . 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 738	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑓 = 𝐹)
32	31	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘𝐹))
33	32	oveqd 7429	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
34	33	dmeqd 5897	. . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
37	15	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
38		simplr 780	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
39		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
40	10, 35, 36, 37, 38, 39	funcf2 17926	. . . . . . . . . . 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 6718	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝐹)𝑦) = (𝑥𝐻𝑦))
42	34, 41	eqtrd 2798	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → dom (𝑥(2nd ‘𝑓)𝑦) = (𝑥𝐻𝑦))
43	33	fveq1d 6885	. . . . . . . . . 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 738	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑔 = 𝐺)
45	44	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑔) = (2nd ‘𝐺))
46	45	oveqd 7429	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑔)𝑦) = (𝑥(2nd ‘𝐺)𝑦))
47	46	fveq1d 6885	. . . . . . . . . 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 4847	. . . . . . . . 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 5199	. . . . . . . 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 1127	. . . . . . 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 7489	. . . . . 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 2798	. . . . 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 4847	. . . 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 3891	. . 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 3478	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
56		prfval.d	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
57	56	elexd 3478	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
58		opex 5447	. . . 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 7564	. 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 2810	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 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⦋csb 3854  ⟨cop 4596   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  Rel wrel 5668  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322   Func cfunc 17912   ⟨,⟩_F cprf 18228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ixp 8897  df-func 17916  df-prf 18232

This theorem is referenced by:  prf1  18257  prf2fval  18258  prfcl  18260  prf1st  18261  prf2nd  18262  1st2ndprf  18263