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Theorem funcestrcsetclem7 17101
Description: Lemma 7 for funcestrcsetc 17104. (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 2799 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
91, 2, 3, 4, 5, 6, 7, 8, 8funcestrcsetclem5 17099 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑋𝐵)) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋))))
109anabsan2 665 . . 3 ((𝜑𝑋𝐵) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋))))
11 eqid 2799 . . . 4 (Id‘𝐸) = (Id‘𝐸)
125adantr 473 . . . 4 ((𝜑𝑋𝐵) → 𝑈 ∈ WUni)
131, 5estrcbas 17079 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
1413, 3syl6reqr 2852 . . . . . 6 (𝜑𝐵 = 𝑈)
1514eleq2d 2864 . . . . 5 (𝜑 → (𝑋𝐵𝑋𝑈))
1615biimpa 469 . . . 4 ((𝜑𝑋𝐵) → 𝑋𝑈)
171, 11, 12, 16estrcid 17088 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝐸)‘𝑋) = ( I ↾ (Base‘𝑋)))
1810, 17fveq12d 6418 . 2 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = (( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))‘( I ↾ (Base‘𝑋))))
19 fvex 6424 . . . . 5 (Base‘𝑋) ∈ V
2019, 19pm3.2i 463 . . . 4 ((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V)
2120a1i 11 . . 3 ((𝜑𝑋𝐵) → ((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))
2422, 23ax-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)))
2624, 25mpbiri 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)))
2821, 26, 273syl 18 . 2 ((𝜑𝑋𝐵) → (( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋)))
291, 2, 3, 4, 5, 6funcestrcsetclem1 17095 . . . 4 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
3029fveq2d 6415 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(𝐹𝑋)) = ((Id‘𝑆)‘(Base‘𝑋)))
31 eqid 2799 . . . 4 (Id‘𝑆) = (Id‘𝑆)
321, 3, 5estrcbasbas 17085 . . . 4 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ 𝑈)
332, 31, 12, 32setcid 17050 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋)))
3430, 33eqtr2d 2834 . 2 ((𝜑𝑋𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
3518, 28, 343eqtrd 2837 1 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
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-ontowf1o 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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