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Theorem inclfusubc 17868

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 17834	. . 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 2729	. . . . 5 ⊢ (id_func‘𝑆) = (id_func‘𝑆)
5	3, 4	rescfth 17864	. . . 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 3935	. 2 ⊢ (𝜑 → (id_func‘𝑆) ∈ (𝑆 Func 𝐶))
8		df-br 5096	. . 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 4835	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)
12		inclfusubc.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	3, 4, 12	idfusubc 17825	. . . . . 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 2767	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = (id_func‘𝑆))
16	15	eleq1d 2813	. . 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 2109 ⟨cop 4585 class class class wbr 5095 I cid 5517 ↾ cres 5625 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 Basecbs 17138 ↾_cat cresc 17733 Subcatcsubc 17734 Func cfunc 17779 id_funccidfu 17780 Faith cfth 17830

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-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-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 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-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-hom 17203 df-cco 17204 df-cat 17592 df-cid 17593 df-homf 17594 df-ssc 17735 df-resc 17736 df-subc 17737 df-func 17783 df-idfu 17784 df-full 17831 df-fth 17832

This theorem is referenced by: rngcifuestrc 20542

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