Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcestrcsetclem7 Structured version   Visualization version   GIF version

Theorem funcestrcsetclem7 16780
 Description: Lemma 7 for funcestrcsetc 16783. (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 2621 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
91, 2, 3, 4, 5, 6, 7, 8, 8funcestrcsetclem5 16778 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑋𝐵)) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋))))
109anabsan2 863 . . 3 ((𝜑𝑋𝐵) → (𝑋𝐺𝑋) = ( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋))))
11 eqid 2621 . . . 4 (Id‘𝐸) = (Id‘𝐸)
125adantr 481 . . . 4 ((𝜑𝑋𝐵) → 𝑈 ∈ WUni)
131, 5estrcbas 16759 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
1413, 3syl6reqr 2674 . . . . . 6 (𝜑𝐵 = 𝑈)
1514eleq2d 2686 . . . . 5 (𝜑 → (𝑋𝐵𝑋𝑈))
1615biimpa 501 . . . 4 ((𝜑𝑋𝐵) → 𝑋𝑈)
171, 11, 12, 16estrcid 16768 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝐸)‘𝑋) = ( I ↾ (Base‘𝑋)))
1810, 17fveq12d 6195 . 2 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = (( I ↾ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))‘( I ↾ (Base‘𝑋))))
19 fvex 6199 . . . . 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 6172 . . . . 5 ( I ↾ (Base‘𝑋)):(Base‘𝑋)–1-1-onto→(Base‘𝑋)
23 f1of 6135 . . . . 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 7867 . . . 4 (((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) → (( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)) ↔ ( I ↾ (Base‘𝑋)):(Base‘𝑋)⟶(Base‘𝑋)))
2624, 25mpbiri 248 . . 3 (((Base‘𝑋) ∈ V ∧ (Base‘𝑋) ∈ V) → ( I ↾ (Base‘𝑋)) ∈ ((Base‘𝑋) ↑𝑚 (Base‘𝑋)))
27 fvresi 6436 . . 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 16774 . . . 4 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
3029fveq2d 6193 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(𝐹𝑋)) = ((Id‘𝑆)‘(Base‘𝑋)))
31 eqid 2621 . . . 4 (Id‘𝑆) = (Id‘𝑆)
321, 3, 5estrcbasbas 16765 . . . 4 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ 𝑈)
332, 31, 12, 32setcid 16730 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋)))
3430, 33eqtr2d 2656 . 2 ((𝜑𝑋𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
3518, 28, 343eqtrd 2659 1 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989  Vcvv 3198   ↦ cmpt 4727   I cid 5021   ↾ cres 5114  ⟶wf 5882  –1-1-onto→wf1o 5885  ‘cfv 5886  (class class class)co 6647   ↦ cmpt2 6649   ↑𝑚 cmap 7854  WUnicwun 9519  Basecbs 15851  Idccid 16320  SetCatcsetc 16719  ExtStrCatcestrc 16756 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-wun 9521  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-z 11375  df-dec 11491  df-uz 11685  df-fz 12324  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-hom 15960  df-cco 15961  df-cat 16323  df-cid 16324  df-setc 16720  df-estrc 16757 This theorem is referenced by:  funcestrcsetc  16783
 Copyright terms: Public domain W3C validator