idfu2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > idfu2		Structured version Visualization version GIF version

Theorem idfu2 17968

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 17967	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
8	7	fveq1d 6881	. 2 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹))
9		idfu2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
10		fvresi 7172	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
11	9, 10	syl 18	. 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹)
12	8, 11	eqtrd 2795	1 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐼)𝑌)‘𝐹) = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 I cid 5549 ↾ cres 5657 ‘cfv 6533 (class class class)co 7414 2^nd c2nd 7986 Basecbs 17302 Hom chom 17354 Catccat 17753 id_funccidfu 17945

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-2nd 7988 df-idfu 17949

This theorem is used by: idfucl 17971 cofid2a 50040

W3C validator