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Theorem prf1 18125

Description: Value of the pairing functor on objects. (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))
prf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
prf1	⊢ (𝜑 → ((1^st ‘𝑃)‘𝑋) = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)

Proof of Theorem prf1 Dummy variables ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfval.k	. . . 4 ⊢ 𝑃 = (𝐹 ⟨,⟩_F 𝐺)
2		prfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		prfval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4		prfval.c	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		prfval.d	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
6	1, 2, 3, 4, 5	prfval 18124	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
7	2	fvexi 6848	. . . . 5 ⊢ 𝐵 ∈ V
8	7	mptex 7169	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩) ∈ V
9	7, 7	mpoex 8023	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V
10	8, 9	op1std 7943	. . 3 ⊢ (𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ → (1^st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩))
11	6, 10	syl 17	. 2 ⊢ (𝜑 → (1^st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩))
12		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
13	12	fveq2d 6838	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
14	12	fveq2d 6838	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
15	13, 14	opeq12d 4837	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
16		prf1.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		opex 5412	. . 3 ⊢ ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ ∈ V
18	17	a1i 11	. 2 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ ∈ V)
19	11, 15, 16, 18	fvmptd 6948	1 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑋) = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ⟨cop 4586 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 1^st c1st 7931 2^nd c2nd 7932 Basecbs 17138 Hom chom 17190 Func cfunc 17780 ⟨,⟩_F cprf 18096

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8767 df-ixp 8838 df-func 17784 df-prf 18100

This theorem is referenced by: prfcl 18128 uncf1 18161 uncf2 18162 yonedalem21 18198 yonedalem22 18203

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