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Theorem estrchomfeqhom 18069

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 18068	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
5	1, 2	estrcbas 18058	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
6	5	eqcomd 2737	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = 𝑈)
7	6	sqxpeqd 5701	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = (𝑈 × 𝑈))
8	7	fneq2d 6632	. . 3 ⊢ (𝜑 → (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐻 Fn (𝑈 × 𝑈)))
9	4, 8	mpbird 256	. 2 ⊢ (𝜑 → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		eqid 2731	. . 3 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘𝐶)
11		eqid 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	10, 11, 3	fnhomeqhomf 17617	. 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 1541 ∈ wcel 2106 × cxp 5667 Fn wfn 6527 ‘cfv 6532 Basecbs 17126 Hom chom 17190 Hom_f chomf 17592 ExtStrCatcestrc 18055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-slot 17097 df-ndx 17109 df-base 17127 df-hom 17203 df-cco 17204 df-homf 17596 df-estrc 18056

This theorem is referenced by: rnghmsubcsetc 46521 rhmsubcsetc 46567

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