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Theorem fucid 18063

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 17952	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	5	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		fucid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
9	2, 6, 7, 8	fuccatid 18061	. . . 4 ⊢ (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓)))))
10	9	simprd 501	. . 3 ⊢ (𝜑 → (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓))))
11	1, 10	eqtrid 2807	. 2 ⊢ (𝜑 → 𝐼 = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1^st ‘𝑓))))
12		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
13	12	fveq2d 6882	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (1^st ‘𝑓) = (1^st ‘𝐹))
14	13	coeq2d 5842	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ( 1 ∘ (1^st ‘𝑓)) = ( 1 ∘ (1^st ‘𝐹)))
15	8	fvexi 6892	. . . 4 ⊢ 1 ∈ V
16		fvex 6891	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
17	15, 16	coex 7927	. . 3 ⊢ ( 1 ∘ (1^st ‘𝐹)) ∈ V
18	17	a1i 11	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ V)
19	11, 14, 3, 18	fvmptd 6994	1 ⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1^st ‘𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ↦ cmpt 5186 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7413 1^st c1st 7984 Catccat 17752 Idccid 17753 Func cfunc 17943 FuncCat cfuc 18034

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 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 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-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 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-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 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-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 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-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-slot 17274 df-ndx 17286 df-base 17302 df-hom 17366 df-cco 17367 df-cat 17756 df-cid 17757 df-func 17947 df-nat 18035 df-fuc 18036

This theorem is used by: fucsect 18064 evlfcl 18310 curfcl 18320 curfuncf 18326 curf2ndf 18335 fuco11id 50260 fucoid 50274 fucolid 50287 fucorid 50288 precofvalALT 50294 fucoppcid 50334

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