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Theorem estrccofval 18090

Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)

**Hypotheses**
Ref	Expression
estrcbas.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
estrcbas.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
estrcco.o	⊢ · = (comp‘𝐶)

**Assertion**
Ref	Expression
estrccofval	⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))

Distinct variable groups: 𝑓,𝑔,𝑣,𝑧 𝜑,𝑣,𝑧 𝑣,𝑈,𝑧 Allowed substitution hints: 𝜑(𝑓,𝑔) 𝐶(𝑧,𝑣,𝑓,𝑔) · (𝑧,𝑣,𝑓,𝑔) 𝑈(𝑓,𝑔) 𝑉(𝑧,𝑣,𝑓,𝑔)

Proof of Theorem estrccofval Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		estrcbas.c	. . 3 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		estrcbas.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqid 2741	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	estrchomfval 18087	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
5		eqidd 2742	. . 3 ⊢ (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
6	1, 2, 4, 5	estrcval 18085	. 2 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
7		catstr 17922	. 2 ⊢ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩} Struct ⟨1, ;15⟩
8		ccoid 17372	. 2 ⊢ comp = Slot (comp‘ndx)
9		snsstp3 4752	. 2 ⊢ {⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩} ⊆ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩}
10	2, 2	xpexd 7698	. . 3 ⊢ (𝜑 → (𝑈 × 𝑈) ∈ V)
11		mpoexga 8023	. . 3 ⊢ (((𝑈 × 𝑈) ∈ V ∧ 𝑈 ∈ 𝑉) → (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) ∈ V)
12	10, 2, 11	syl2anc 591	. 2 ⊢ (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) ∈ V)
13		estrcco.o	. 2 ⊢ · = (comp‘𝐶)
14	6, 7, 8, 9, 12, 13	strfv3 17169	1 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1548 ∈ wcel 2121 Vcvv 3433 {ctp 4562 ⟨cop 4564 × cxp 5619 ∘ ccom 5625 ‘cfv 6489 (class class class)co 7360 ∈ cmpo 7362 1^st c1st 7933 2^nd c2nd 7934 ↑_m cmap 8767 1c1 11034 5c5 12234 cdc 12639 ndxcnx 17158 Basecbs 17174 Hom chom 17226 compcco 17227 ExtStrCatcestrc 18083

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 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 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-estrc 18084

This theorem is referenced by: estrcco 18091 dfrngc2 20604 dfringc2 20633

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