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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcsetcestrc 17406. (Contributed by AV, 27-Mar-2020.) |
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) |
Ref | Expression |
---|---|
funcsetcestrclem7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
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 17401 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
8 | 7 | anabsan2 673 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
9 | eqid 2798 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
10 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑈 ∈ WUni) |
11 | 1, 4 | setcbas 17330 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
12 | 2, 11 | eqtr4id 2852 | . . . . . 6 ⊢ (𝜑 → 𝐶 = 𝑈) |
13 | 12 | eleq2d 2875 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
14 | 13 | biimpa 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
15 | 1, 9, 10, 14 | setcid 17338 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝑆)‘𝑋) = ( I ↾ 𝑋)) |
16 | 8, 15 | fveq12d 6652 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋))) |
17 | f1oi 6627 | . . . . . 6 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
18 | f1of 6590 | . . . . . 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 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
21 | 20, 20 | elmapd 8403 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋) ↔ ( I ↾ 𝑋):𝑋⟶𝑋)) |
22 | 19, 21 | mpbiri 261 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋)) |
23 | fvresi 6912 | . . . 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 2798 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
26 | 25 | 1strbas 16591 | . . . . 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 5818 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
29 | 24, 28 | eqtrd 2833 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
30 | 1, 2, 3 | funcsetcestrclem1 17396 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
31 | 30 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘(𝐹‘𝑋)) = ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉})) |
32 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
33 | eqid 2798 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
34 | 1, 2, 4, 5 | setc1strwun 17395 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
35 | 32, 33, 10, 34 | estrcid 17376 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉}) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
36 | 31, 35 | eqtr2d 2834 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉})) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
37 | 16, 29, 36 | 3eqtrd 2837 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ωcom 7560 ↑m cmap 8389 WUnicwun 10111 ndxcnx 16472 Basecbs 16475 Idccid 16928 SetCatcsetc 17327 ExtStrCatcestrc 17364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wun 10113 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-plp 10394 df-ltp 10396 df-enr 10466 df-nr 10467 df-c 10532 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-hom 16581 df-cco 16582 df-cat 16931 df-cid 16932 df-setc 17328 df-estrc 17365 |
This theorem is referenced by: funcsetcestrc 17406 |
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