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Theorem isf32lem12 8006
Description: Lemma for isfin3-2 8009. (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 6808 . . . . 5  |-  ( f  e.  ( ~P G  ^m  om )  ->  f : om --> ~P G )
2 isf32lem11 8005 . . . . . . . . . 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 2645 . 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 7971 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165   ~Pcpw 3638   |^|cint 3878   class class class wbr 4039   suc csuc 4410   omcom 4672   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ~<_* cwdom 7287
This theorem is referenced by:  isf33lem  8008  isfin3-2  8009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-wdom 7289  df-card 7588
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