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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcestrcsetc 17387. (Contributed by AV, 23-Mar-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcestrcsetc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcestrcsetc.b | ⊢ 𝐵 = (Base‘𝐸) |
funcestrcsetc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcestrcsetc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcestrcsetc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcestrcsetc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) |
Ref | Expression |
---|---|
funcestrcsetclem7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.e | . . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
2 | funcestrcsetc.s | . . . . 5 ⊢ 𝑆 = (SetCat‘𝑈) | |
3 | funcestrcsetc.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐸) | |
4 | funcestrcsetc.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
5 | funcestrcsetc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcestrcsetc.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) | |
7 | funcestrcsetc.g | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) | |
8 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8, 8 | funcestrcsetclem5 17382 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))) |
10 | 9 | anabsan2 670 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))) |
11 | eqid 2818 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
12 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑈 ∈ WUni) |
13 | 1, 5 | estrcbas 17363 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸)) |
14 | 13, 3 | syl6reqr 2872 | . . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑈) |
15 | 14 | eleq2d 2895 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈)) |
16 | 15 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝑈) |
17 | 1, 11, 12, 16 | estrcid 17372 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝐸)‘𝑋) = ( I ↾ (Base‘𝑋))) |
18 | 10, 17 | fveq12d 6670 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = (( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))‘( I ↾ (Base‘𝑋)))) |
19 | fvex 6676 | . . . . 5 ⊢ (Base‘𝑋) ∈ V | |
20 | 19, 19 | pm3.2i 471 | . . . 4 ⊢ ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V)) |
22 | f1oi 6645 | . . . . 5 ⊢ ( I ↾ (Base‘𝑋)):(Base‘𝑋)–1-1-onto→(Base‘𝑋) | |
23 | f1of 6608 | . . . . 5 ⊢ (( I ↾ (Base‘𝑋)):(Base‘𝑋)–1-1-onto→(Base‘𝑋) → ( I ↾ (Base‘𝑋)):(Base‘𝑋)⟶(Base‘𝑋)) | |
24 | 22, 23 | ax-mp 5 | . . . 4 ⊢ ( I ↾ (Base‘𝑋)):(Base‘𝑋)⟶(Base‘𝑋) |
25 | elmapg 8408 | . . . 4 ⊢ (((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) → (( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑m (Base‘𝑋)) ↔ ( I ↾ (Base‘𝑋)):(Base‘𝑋)⟶(Base‘𝑋))) | |
26 | 24, 25 | mpbiri 259 | . . 3 ⊢ (((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) → ( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑m (Base‘𝑋))) |
27 | fvresi 6927 | . . 3 ⊢ (( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑m (Base‘𝑋)) → (( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋))) | |
28 | 21, 26, 27 | 3syl 18 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋))) |
29 | 1, 2, 3, 4, 5, 6 | funcestrcsetclem1 17378 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
30 | 29 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
31 | eqid 2818 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
32 | 1, 3, 5 | estrcbasbas 17369 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
33 | 2, 31, 12, 32 | setcid 17334 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
34 | 30, 33 | eqtr2d 2854 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
35 | 18, 28, 34 | 3eqtrd 2857 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 I cid 5452 ↾ cres 5550 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 WUnicwun 10110 Basecbs 16471 Idccid 16924 SetCatcsetc 17323 ExtStrCatcestrc 17360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-wun 10112 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-setc 17324 df-estrc 17361 |
This theorem is referenced by: funcestrcsetc 17387 |
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