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Theorem inclfusubc 17956

Description: The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)

**Hypotheses**
Ref	Expression
inclfusubc.j	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
inclfusubc.s	⊢ 𝑆 = (𝐶 ↾_cat 𝐽)
inclfusubc.b	⊢ 𝐵 = (Base‘𝑆)
inclfusubc.f	⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))
inclfusubc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦))))

**Assertion**
Ref	Expression
inclfusubc	⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺)

Distinct variable groups: 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝑥,𝐽,𝑦 𝑥,𝑆,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦)

**Proof of Theorem inclfusubc**
Step	Hyp	Ref	Expression
1		fthfunc 17922	. . 3 ⊢ (𝑆 Faith 𝐶) ⊆ (𝑆 Func 𝐶)
2		inclfusubc.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
3		inclfusubc.s	. . . . 5 ⊢ 𝑆 = (𝐶 ↾_cat 𝐽)
4		eqid 2735	. . . . 5 ⊢ (id_func‘𝑆) = (id_func‘𝑆)
5	3, 4	rescfth 17952	. . . 4 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (id_func‘𝑆) ∈ (𝑆 Faith 𝐶))
6	2, 5	syl 17	. . 3 ⊢ (𝜑 → (id_func‘𝑆) ∈ (𝑆 Faith 𝐶))
7	1, 6	sselid 3956	. 2 ⊢ (𝜑 → (id_func‘𝑆) ∈ (𝑆 Func 𝐶))
8		df-br 5120	. . 3 ⊢ (𝐹(𝑆 Func 𝐶)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆 Func 𝐶))
9		inclfusubc.f	. . . . . 6 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))
10		inclfusubc.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦))))
11	9, 10	opeq12d 4857	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)
12		inclfusubc.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	3, 4, 12	idfusubc 17913	. . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (id_func‘𝑆) = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)
14	2, 13	syl 17	. . . . 5 ⊢ (𝜑 → (id_func‘𝑆) = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)
15	11, 14	eqtr4d 2773	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = (id_func‘𝑆))
16	15	eleq1d 2819	. . 3 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝑆 Func 𝐶) ↔ (id_func‘𝑆) ∈ (𝑆 Func 𝐶)))
17	8, 16	bitrid 283	. 2 ⊢ (𝜑 → (𝐹(𝑆 Func 𝐶)𝐺 ↔ (id_func‘𝑆) ∈ (𝑆 Func 𝐶)))
18	7, 17	mpbird 257	1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 class class class wbr 5119 I cid 5547 ↾ cres 5656 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17228 ↾_cat cresc 17821 Subcatcsubc 17822 Func cfunc 17867 id_funccidfu 17868 Faith cfth 17918

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-hom 17295 df-cco 17296 df-cat 17680 df-cid 17681 df-homf 17682 df-ssc 17823 df-resc 17824 df-subc 17825 df-func 17871 df-idfu 17872 df-full 17919 df-fth 17920

This theorem is referenced by: rngcifuestrc 20599

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