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Theorem isf32lem12 8244
Description: Lemma for isfin3-2 8247. (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 7038 . . . . 5  |-  ( f  e.  ( ~P G  ^m  om )  ->  f : om --> ~P G )
2 isf32lem11 8243 . . . . . . . . . 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 425 . . . . . . . . 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 1153 . . . . . . . 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 428 . . . . . . 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 118 . . . . . 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 588 . . . . 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 16 . . . 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 81 . . 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 2795 . 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 8209 . 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 226 1  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705    C_ wss 3320   ~Pcpw 3799   |^|cint 4050   class class class wbr 4212   suc csuc 4583   omcom 4845   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018    ~<_* cwdom 7525
This theorem is referenced by:  isf33lem  8246  isfin3-2  8247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-1o 6724  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-wdom 7527  df-card 7826
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