idfu2 - Metamath Proof Explorer

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Theorem idfu2 17929

Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)

**Hypotheses**
Ref	Expression
idfuval.i	⊢ 𝐼 = (id_func‘𝐶)
idfuval.b	⊢ 𝐵 = (Base‘𝐶)
idfuval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idfuval.h	⊢ 𝐻 = (Hom ‘𝐶)
idfu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
idfu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
idfu2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

**Assertion**
Ref	Expression
idfu2	⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = 𝐹)

**Proof of Theorem idfu2**
Step	Hyp	Ref	Expression
1		idfuval.i	. . . 4 ⊢ 𝐼 = (id_func‘𝐶)
2		idfuval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		idfuval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		idfuval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
5		idfu2nd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		idfu2nd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	idfu2nd 17928	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
8	7	fveq1d 6909	. 2 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹))
9		idfu2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
10		fvresi 7193	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
12	8, 11	eqtrd 2775	1 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 I cid 5582 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 2^nd c2nd 8012 Basecbs 17245 Hom chom 17309 Catccat 17709 id_funccidfu 17906

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-2nd 8014 df-idfu 17910

This theorem is referenced by: idfucl 17932

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