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Theorem estrchomfeqhom 18096

Description: The functionalized Hom-set operation equals the Hom-set operation in the category of extensible structures (in a universe). (Contributed by AV, 8-Mar-2020.)

**Hypotheses**
Ref	Expression
estrchomfn.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
estrchomfn.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
estrchomfn.h	⊢ 𝐻 = (Hom ‘𝐶)

**Assertion**
Ref	Expression
estrchomfeqhom	⊢ (𝜑 → (Hom_f ‘𝐶) = 𝐻)

**Proof of Theorem estrchomfeqhom**
Step	Hyp	Ref	Expression
1		estrchomfn.c	. . . 4 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		estrchomfn.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		estrchomfn.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	estrchomfn 18095	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
5	1, 2	estrcbas 18085	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
6	5	eqcomd 2742	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = 𝑈)
7	6	sqxpeqd 5653	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = (𝑈 × 𝑈))
8	7	fneq2d 6582	. . 3 ⊢ (𝜑 → (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐻 Fn (𝑈 × 𝑈)))
9	4, 8	mpbird 258	. 2 ⊢ (𝜑 → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		eqid 2736	. . 3 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘𝐶)
11		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	10, 11, 3	fnhomeqhomf 17651	. 2 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (Hom_f ‘𝐶) = 𝐻)
13	9, 12	syl 17	1 ⊢ (𝜑 → (Hom_f ‘𝐶) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2115 × cxp 5619 Fn wfn 6483 ‘cfv 6488 Basecbs 17173 Hom chom 17225 Hom_f chomf 17626 ExtStrCatcestrc 18082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1970 ax-7 2011 ax-8 2117 ax-9 2125 ax-10 2148 ax-11 2164 ax-12 2185 ax-ext 2708 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7681 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 850 df-3or 1089 df-3an 1090 df-tru 1546 df-fal 1556 df-ex 1783 df-nf 1787 df-sb 2070 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3061 df-reu 3342 df-rab 3389 df-v 3430 df-sbc 3727 df-csb 3835 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3906 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7934 df-2nd 7935 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-fz 13456 df-struct 17111 df-slot 17146 df-ndx 17158 df-base 17174 df-hom 17238 df-cco 17239 df-homf 17630 df-estrc 18083

This theorem is referenced by: rnghmsubcsetc 20608 rhmsubcsetc 20637

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