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Theorem estrchomfeqhom 18103

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 18102	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
5	1, 2	estrcbas 18092	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
6	5	eqcomd 2736	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = 𝑈)
7	6	sqxpeqd 5672	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = (𝑈 × 𝑈))
8	7	fneq2d 6614	. . 3 ⊢ (𝜑 → (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐻 Fn (𝑈 × 𝑈)))
9	4, 8	mpbird 257	. 2 ⊢ (𝜑 → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		eqid 2730	. . 3 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘𝐶)
11		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	10, 11, 3	fnhomeqhomf 17658	. 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 1540 ∈ wcel 2109 × cxp 5638 Fn wfn 6508 ‘cfv 6513 Basecbs 17185 Hom chom 17237 Hom_f chomf 17633 ExtStrCatcestrc 18089

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-fz 13475 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-homf 17637 df-estrc 18090

This theorem is referenced by: rnghmsubcsetc 20548 rhmsubcsetc 20577

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