fucid - Metamath Proof Explorer

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

Theorem fucid 17992

Description: The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fucid.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucid.i	⊢ 𝐼 = (Id‘𝑄)
fucid.1	⊢ 1 = (Id‘𝐷)
fucid.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
fucid	⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1^st ‘𝐹)))

Proof of Theorem fucid Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucid.i	. . 3 ⊢ 𝐼 = (Id‘𝑄)
2		fucid.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
3		fucid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 17881	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	5	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		fucid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
9	2, 6, 7, 8	fuccatid 17990	. . . 4 ⊢ (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓)))))
10	9	simprd 495	. . 3 ⊢ (𝜑 → (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓))))
11	1, 10	eqtrid 2783	. 2 ⊢ (𝜑 → 𝐼 = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓))))
12		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
13	12	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (1^st ‘𝑓) = (1^st ‘𝐹))
14	13	coeq2d 5847	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ( 1 ∘ (1^st ‘𝑓)) = ( 1 ∘ (1^st ‘𝐹)))
15	8	fvexi 6895	. . . 4 ⊢ 1 ∈ V
16		fvex 6894	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
17	15, 16	coex 7931	. . 3 ⊢ ( 1 ∘ (1^st ‘𝐹)) ∈ V
18	17	a1i 11	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ V)
19	11, 14, 3, 18	fvmptd 6998	1 ⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1^st ‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ↦ cmpt 5206 ∘ ccom 5663 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 Catccat 17681 Idccid 17682 Func cfunc 17872 FuncCat cfuc 17963

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-hom 17300 df-cco 17301 df-cat 17685 df-cid 17686 df-func 17876 df-nat 17964 df-fuc 17965

This theorem is referenced by: fucsect 17993 evlfcl 18239 curfcl 18249 curfuncf 18255 curf2ndf 18264 fuco11id 49212 fucoid 49226 fucolid 49239 fucorid 49240 precofvalALT 49246

W3C validator