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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcestrcsetc 17104. (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 2799 | . . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8, 8 | funcestrcsetclem5 17099 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))) |
10 | 9 | anabsan2 665 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))) |
11 | eqid 2799 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
12 | 5 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑈 ∈ WUni) |
13 | 1, 5 | estrcbas 17079 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸)) |
14 | 13, 3 | syl6reqr 2852 | . . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑈) |
15 | 14 | eleq2d 2864 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈)) |
16 | 15 | biimpa 469 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝑈) |
17 | 1, 11, 12, 16 | estrcid 17088 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝐸)‘𝑋) = ( I ↾ (Base‘𝑋))) |
18 | 10, 17 | fveq12d 6418 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = (( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))‘( I ↾ (Base‘𝑋)))) |
19 | fvex 6424 | . . . . 5 ⊢ (Base‘𝑋) ∈ V | |
20 | 19, 19 | pm3.2i 463 | . . . 4 ⊢ ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V)) |
22 | f1oi 6393 | . . . . 5 ⊢ ( I ↾ (Base‘𝑋)):(Base‘𝑋)–1-1-onto→(Base‘𝑋) | |
23 | f1of 6356 | . . . . 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 8108 | . . . 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 250 | . . 3 ⊢ (((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) → ( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑𝑚 (Base‘𝑋))) |
27 | fvresi 6668 | . . 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 17095 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
30 | 29 | fveq2d 6415 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
31 | eqid 2799 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
32 | 1, 3, 5 | estrcbasbas 17085 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
33 | 2, 31, 12, 32 | setcid 17050 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
34 | 30, 33 | eqtr2d 2834 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
35 | 18, 28, 34 | 3eqtrd 2837 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ↦ cmpt 4922 I cid 5219 ↾ cres 5314 ⟶wf 6097 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ↑𝑚 cmap 8095 WUnicwun 9810 Basecbs 16184 Idccid 16640 SetCatcsetc 17039 ExtStrCatcestrc 17076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-wun 9812 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-hom 16291 df-cco 16292 df-cat 16643 df-cid 16644 df-setc 17040 df-estrc 17077 |
This theorem is referenced by: funcestrcsetc 17104 |
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