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Theorem fucid 17899

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 17788	. . . . . . 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 17897	. . . 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 2776	. 2 ⊢ (𝜑 → 𝐼 = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓))))
12		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
13	12	fveq2d 6830	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (1^st ‘𝑓) = (1^st ‘𝐹))
14	13	coeq2d 5809	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ( 1 ∘ (1^st ‘𝑓)) = ( 1 ∘ (1^st ‘𝐹)))
15	8	fvexi 6840	. . . 4 ⊢ 1 ∈ V
16		fvex 6839	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
17	15, 16	coex 7870	. . 3 ⊢ ( 1 ∘ (1^st ‘𝐹)) ∈ V
18	17	a1i 11	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ V)
19	11, 14, 3, 18	fvmptd 6941	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 3438 ↦ cmpt 5176 ∘ ccom 5627 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 Catccat 17588 Idccid 17589 Func cfunc 17779 FuncCat cfuc 17870

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-hom 17203 df-cco 17204 df-cat 17592 df-cid 17593 df-func 17783 df-nat 17871 df-fuc 17872

This theorem is referenced by: fucsect 17900 evlfcl 18146 curfcl 18156 curfuncf 18162 curf2ndf 18171 fuco11id 49320 fucoid 49334 fucolid 49347 fucorid 49348 precofvalALT 49354 fucoppcid 49394

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