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Theorem funcsetcestrclem7 18129

Description: Lemma 7 for funcsetcestrc 18132. (Contributed by AV, 27-Mar-2020.)

**Hypotheses**
Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcsetcestrc.o	⊢ (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
funcsetcestrc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)

**Assertion**
Ref	Expression
funcsetcestrclem7	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))

Distinct variable groups: 𝑥,𝐶 𝑥,𝑋 𝜑,𝑥 𝑦,𝐶,𝑥 𝑦,𝑋 𝜑,𝑦 Allowed substitution hints: 𝑆(𝑥,𝑦) 𝑈(𝑥,𝑦) 𝐸(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦)

**Proof of Theorem funcsetcestrclem7**
Step	Hyp	Ref	Expression
1		funcsetcestrc.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
2		funcsetcestrc.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑆)
3		funcsetcestrc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
4		funcsetcestrc.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni)
5		funcsetcestrc.o	. . . . 5 ⊢ (𝜑 → ω ∈ 𝑈)
6		funcsetcestrc.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
7	1, 2, 3, 4, 5, 6	funcsetcestrclem5 18127	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑_m 𝑋)))
8	7	anabsan2 674	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑_m 𝑋)))
9		eqid 2730	. . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆)
10	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑈 ∈ WUni)
11	1, 4	setcbas 18047	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
12	2, 11	eqtr4id 2784	. . . . . 6 ⊢ (𝜑 → 𝐶 = 𝑈)
13	12	eleq2d 2815	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈))
14	13	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈)
15	1, 9, 10, 14	setcid 18055	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝑆)‘𝑋) = ( I ↾ 𝑋))
16	8, 15	fveq12d 6868	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = (( I ↾ (𝑋 ↑_m 𝑋))‘( I ↾ 𝑋)))
17		f1oi 6841	. . . . . 6 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
18		f1of 6803	. . . . . 6 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋⟶𝑋)
19	17, 18	ax-mp 5	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋⟶𝑋
20		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
21	20, 20	elmapd 8816	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ 𝑋) ∈ (𝑋 ↑_m 𝑋) ↔ ( I ↾ 𝑋):𝑋⟶𝑋))
22	19, 21	mpbiri 258	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) ∈ (𝑋 ↑_m 𝑋))
23		fvresi 7150	. . . 4 ⊢ (( I ↾ 𝑋) ∈ (𝑋 ↑_m 𝑋) → (( I ↾ (𝑋 ↑_m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ 𝑋))
24	22, 23	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑_m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ 𝑋))
25		eqid 2730	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝑋⟩} = {⟨(Base‘ndx), 𝑋⟩}
26	25	1strbas 17201	. . . . 5 ⊢ (𝑋 ∈ 𝐶 → 𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
27	20, 26	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
28	27	reseq2d 5953	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) = ( I ↾ (Base‘{⟨(Base‘ndx), 𝑋⟩})))
29	24, 28	eqtrd 2765	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑_m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ (Base‘{⟨(Base‘ndx), 𝑋⟩})))
30	1, 2, 3	funcsetcestrclem1 18122	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})
31	30	fveq2d 6865	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘(𝐹‘𝑋)) = ((Id‘𝐸)‘{⟨(Base‘ndx), 𝑋⟩}))
32		funcsetcestrc.e	. . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈)
33		eqid 2730	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
34	1, 2, 4, 5	setc1strwun 18121	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ 𝑈)
35	32, 33, 10, 34	estrcid 18102	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘{⟨(Base‘ndx), 𝑋⟩}) = ( I ↾ (Base‘{⟨(Base‘ndx), 𝑋⟩})))
36	31, 35	eqtr2d 2766	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ (Base‘{⟨(Base‘ndx), 𝑋⟩})) = ((Id‘𝐸)‘(𝐹‘𝑋)))
37	16, 29, 36	3eqtrd 2769	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {csn 4592 ⟨cop 4598 ↦ cmpt 5191 I cid 5535 ↾ cres 5643 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ωcom 7845 ↑_m cmap 8802 WUnicwun 10660 ndxcnx 17170 Basecbs 17186 Idccid 17633 SetCatcsetc 18044 ExtStrCatcestrc 18090

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 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-omul 8442 df-er 8674 df-ec 8676 df-qs 8680 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-wun 10662 df-ni 10832 df-pli 10833 df-mi 10834 df-lti 10835 df-plpq 10868 df-mpq 10869 df-ltpq 10870 df-enq 10871 df-nq 10872 df-erq 10873 df-plq 10874 df-mq 10875 df-1nq 10876 df-rq 10877 df-ltnq 10878 df-np 10941 df-plp 10943 df-ltp 10945 df-enr 11015 df-nr 11016 df-c 11081 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-hom 17251 df-cco 17252 df-cat 17636 df-cid 17637 df-setc 18045 df-estrc 18091

This theorem is referenced by: funcsetcestrc 18132

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