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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcsetcestrc 17417. (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 17412 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
8 | 7 | anabsan2 672 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
9 | eqid 2824 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
10 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑈 ∈ WUni) |
11 | 1, 4 | setcbas 17341 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
12 | 11, 2 | syl6reqr 2878 | . . . . . 6 ⊢ (𝜑 → 𝐶 = 𝑈) |
13 | 12 | eleq2d 2901 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
14 | 13 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
15 | 1, 9, 10, 14 | setcid 17349 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝑆)‘𝑋) = ( I ↾ 𝑋)) |
16 | 8, 15 | fveq12d 6680 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋))) |
17 | f1oi 6655 | . . . . . 6 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
18 | f1of 6618 | . . . . . 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 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
21 | 20, 20 | elmapd 8423 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋) ↔ ( I ↾ 𝑋):𝑋⟶𝑋)) |
22 | 19, 21 | mpbiri 260 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋)) |
23 | fvresi 6938 | . . . 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 2824 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
26 | 25 | 1strbas 16602 | . . . . 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 5856 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
29 | 24, 28 | eqtrd 2859 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
30 | 1, 2, 3 | funcsetcestrclem1 17407 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
31 | 30 | fveq2d 6677 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘(𝐹‘𝑋)) = ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉})) |
32 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
33 | eqid 2824 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
34 | 1, 2, 4, 5 | setc1strwun 17406 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
35 | 32, 33, 10, 34 | estrcid 17387 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉}) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
36 | 31, 35 | eqtr2d 2860 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉})) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
37 | 16, 29, 36 | 3eqtrd 2863 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4570 〈cop 4576 ↦ cmpt 5149 I cid 5462 ↾ cres 5560 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 ωcom 7583 ↑m cmap 8409 WUnicwun 10125 ndxcnx 16483 Basecbs 16486 Idccid 16939 SetCatcsetc 17338 ExtStrCatcestrc 17375 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-wun 10127 df-ni 10297 df-pli 10298 df-mi 10299 df-lti 10300 df-plpq 10333 df-mpq 10334 df-ltpq 10335 df-enq 10336 df-nq 10337 df-erq 10338 df-plq 10339 df-mq 10340 df-1nq 10341 df-rq 10342 df-ltnq 10343 df-np 10406 df-plp 10408 df-ltp 10410 df-enr 10480 df-nr 10481 df-c 10546 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-hom 16592 df-cco 16593 df-cat 16942 df-cid 16943 df-setc 17339 df-estrc 17376 |
This theorem is referenced by: funcsetcestrc 17417 |
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