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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcsetcestrc 17416. (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 17411 | . . . 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 2823 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
10 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑈 ∈ WUni) |
11 | 1, 4 | setcbas 17340 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
12 | 11, 2 | syl6reqr 2877 | . . . . . 6 ⊢ (𝜑 → 𝐶 = 𝑈) |
13 | 12 | eleq2d 2900 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
14 | 13 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
15 | 1, 9, 10, 14 | setcid 17348 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝑆)‘𝑋) = ( I ↾ 𝑋)) |
16 | 8, 15 | fveq12d 6679 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋))) |
17 | f1oi 6654 | . . . . . 6 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
18 | f1of 6617 | . . . . . 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 8422 | . . . . 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 6937 | . . . 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 2823 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
26 | 25 | 1strbas 16601 | . . . . 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 5855 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
29 | 24, 28 | eqtrd 2858 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
30 | 1, 2, 3 | funcsetcestrclem1 17406 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
31 | 30 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘(𝐹‘𝑋)) = ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉})) |
32 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
33 | eqid 2823 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
34 | 1, 2, 4, 5 | setc1strwun 17405 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
35 | 32, 33, 10, 34 | estrcid 17386 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉}) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
36 | 31, 35 | eqtr2d 2859 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉})) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
37 | 16, 29, 36 | 3eqtrd 2862 | 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 1537 ∈ wcel 2114 {csn 4569 〈cop 4575 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ωcom 7582 ↑m cmap 8408 WUnicwun 10124 ndxcnx 16482 Basecbs 16485 Idccid 16938 SetCatcsetc 17337 ExtStrCatcestrc 17374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-wun 10126 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-plp 10407 df-ltp 10409 df-enr 10479 df-nr 10480 df-c 10545 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-setc 17338 df-estrc 17375 |
This theorem is referenced by: funcsetcestrc 17416 |
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