idfu2 - Metamath Proof Explorer

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Theorem idfu2 17847

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 17846	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
8	7	fveq1d 6863	. 2 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹))
9		idfu2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
10		fvresi 7150	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
12	8, 11	eqtrd 2765	1 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 I cid 5535 ↾ cres 5643 ‘cfv 6514 (class class class)co 7390 2^nd c2nd 7970 Basecbs 17186 Hom chom 17238 Catccat 17632 id_funccidfu 17824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-2nd 7972 df-idfu 17828

This theorem is referenced by: idfucl 17850 cofid2a 49106

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