idfu2 - Metamath Proof Explorer

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Theorem idfu2 16982

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 16981	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
8	7	fveq1d 6545	. 2 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹))
9		idfu2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
10		fvresi 6803	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
12	8, 11	eqtrd 2831	1 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 I cid 5352 ↾ cres 5450 ‘cfv 6230 (class class class)co 7021 2^nd c2nd 7549 Basecbs 16317 Hom chom 16410 Catccat 16769 id_funccidfu 16959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-ov 7024 df-2nd 7551 df-idfu 16963

This theorem is referenced by: idfucl 16985

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