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Theorem estrchomfeqhom 16823

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 16822	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
5	1, 2	estrcbas 16812	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
6	5	eqcomd 2657	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = 𝑈)
7	6	sqxpeqd 5175	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = (𝑈 × 𝑈))
8	7	fneq2d 6020	. . 3 ⊢ (𝜑 → (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐻 Fn (𝑈 × 𝑈)))
9	4, 8	mpbird 247	. 2 ⊢ (𝜑 → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		eqid 2651	. . 3 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘𝐶)
11		eqid 2651	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	10, 11, 3	fnhomeqhomf 16398	. 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 1523 ∈ wcel 2030 × cxp 5141 Fn wfn 5921 ‘cfv 5926 Basecbs 15904 Hom chom 15999 Hom_f chomf 16374 ExtStrCatcestrc 16809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-hom 16013 df-cco 16014 df-homf 16378 df-estrc 16810

This theorem is referenced by: rnghmsubcsetc 42302 rhmsubcsetc 42348

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