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Theorem prf1 17449

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 17448	. . 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 6683	. . . . 5 ⊢ 𝐵 ∈ V
8	7	mptex 6985	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩) ∈ V
9	7, 7	mpoex 7776	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V
10	8, 9	op1std 7698	. . 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 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
13	12	fveq2d 6673	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
14	12	fveq2d 6673	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
15	13, 14	opeq12d 4810	. 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 5355	. . 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 6774	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 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⟨cop 4572 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1^st c1st 7686 2^nd c2nd 7687 Basecbs 16482 Hom chom 16575 Func cfunc 17123 ⟨,⟩_F cprf 17420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 df-ixp 8461 df-func 17127 df-prf 17424

This theorem is referenced by: prfcl 17452 uncf1 17485 uncf2 17486 yonedalem21 17522 yonedalem22 17527

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