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Theorem funcestrcsetclem7 17384
Description: Lemma 7 for funcestrcsetc 17387. (Contributed by AV, 23-Mar-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b 𝐵 = (Base‘𝐸)
funcestrcsetc.c 𝐶 = (Base‘𝑆)
funcestrcsetc.u (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
Assertion
Ref Expression
funcestrcsetclem7 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcestrcsetclem7
StepHypRef 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 (𝜑𝑈 ∈ WUni)
6 funcestrcsetc.f . . . . 5 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcestrcsetc.g . . . . 5 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
8 eqid 2818 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
91, 2, 3, 4, 5, 6, 7, 8, 8funcestrcsetclem5 17382 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑋𝐵)) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋))))
109anabsan2 670 . . 3 ((𝜑𝑋𝐵) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋))))
11 eqid 2818 . . . 4 (Id‘𝐸) = (Id‘𝐸)
125adantr 481 . . . 4 ((𝜑𝑋𝐵) → 𝑈 ∈ WUni)
131, 5estrcbas 17363 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
1413, 3syl6reqr 2872 . . . . . 6 (𝜑𝐵 = 𝑈)
1514eleq2d 2895 . . . . 5 (𝜑 → (𝑋𝐵𝑋𝑈))
1615biimpa 477 . . . 4 ((𝜑𝑋𝐵) → 𝑋𝑈)
171, 11, 12, 16estrcid 17372 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝐸)‘𝑋) = ( I ↾ (Base‘𝑋)))
1810, 17fveq12d 6670 . 2 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = (( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))‘( I ↾ (Base‘𝑋))))
19 fvex 6676 . . . . 5 (Base‘𝑋) ∈ V
2019, 19pm3.2i 471 . . . 4 ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V)
2120a1i 11 . . 3 ((𝜑𝑋𝐵) → ((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))
2422, 23ax-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)))
2624, 25mpbiri 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)))
2821, 26, 273syl 18 . 2 ((𝜑𝑋𝐵) → (( I ↾ ((Base‘𝑋) ↑m (Base‘𝑋)))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋)))
291, 2, 3, 4, 5, 6funcestrcsetclem1 17378 . . . 4 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
3029fveq2d 6667 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(𝐹𝑋)) = ((Id‘𝑆)‘(Base‘𝑋)))
31 eqid 2818 . . . 4 (Id‘𝑆) = (Id‘𝑆)
321, 3, 5estrcbasbas 17369 . . . 4 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ 𝑈)
332, 31, 12, 32setcid 17334 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋)))
3430, 33eqtr2d 2854 . 2 ((𝜑𝑋𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
3518, 28, 343eqtrd 2857 1 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
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-ontowf1o 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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