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Theorem isf32lem12 7990
Description: Lemma for isfin3-2 7993. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
isf32lem40.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
Assertion
Ref Expression
isf32lem12  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
Distinct variable groups:    g, F    g, a, x, G
Allowed substitution hints:    F( x, a)    V( x, g, a)

Proof of Theorem isf32lem12
Dummy variables  b 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 6792 . . . . 5  |-  ( f  e.  ( ~P G  ^m  om )  ->  f : om --> ~P G )
2 isf32lem11 7989 . . . . . . . . . 10  |-  ( ( G  e.  V  /\  ( f : om --> ~P G  /\  A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  /\  -.  |^| ran  f  e. 
ran  f ) )  ->  om  ~<_*  G )
32expcom 424 . . . . . . . . 9  |-  ( ( f : om --> ~P G  /\  A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  /\  -.  |^| ran  f  e. 
ran  f )  -> 
( G  e.  V  ->  om  ~<_*  G ) )
433expa 1151 . . . . . . . 8  |-  ( ( ( f : om --> ~P G  /\  A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )
)  /\  -.  |^| ran  f  e.  ran  f )  ->  ( G  e.  V  ->  om  ~<_*  G ) )
54impancom 427 . . . . . . 7  |-  ( ( ( f : om --> ~P G  /\  A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )
)  /\  G  e.  V )  ->  ( -.  |^| ran  f  e. 
ran  f  ->  om  ~<_*  G ) )
65con1d 116 . . . . . 6  |-  ( ( ( f : om --> ~P G  /\  A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )
)  /\  G  e.  V )  ->  ( -.  om  ~<_*  G  ->  |^| ran  f  e.  ran  f ) )
76exp31 587 . . . . 5  |-  ( f : om --> ~P G  ->  ( A. b  e. 
om  ( f `  suc  b )  C_  (
f `  b )  ->  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  |^| ran  f  e.  ran  f ) ) ) )
81, 7syl 15 . . . 4  |-  ( f  e.  ( ~P G  ^m  om )  ->  ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  |^| ran  f  e.  ran  f ) ) ) )
98com4t 79 . . 3  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  ( f  e.  ( ~P G  ^m  om )  ->  ( A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b )  ->  |^| ran  f  e.  ran  f ) ) ) )
109ralrimdv 2632 . 2  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  A. f  e.  ( ~P G  ^m  om ) ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) ) )
11 isf32lem40.f . . 3  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
1211isfin3ds 7955 . 2  |-  ( G  e.  V  ->  ( G  e.  F  <->  A. f  e.  ( ~P G  ^m  om ) ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) ) )
1310, 12sylibrd 225 1  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   class class class wbr 4023   suc csuc 4394   omcom 4656   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ~<_* cwdom 7271
This theorem is referenced by:  isf33lem  7992  isfin3-2  7993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wdom 7273  df-card 7572
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