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Theorem isf32lem12 8740
Description: Lemma for isfin3-2 8743. (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 7437 . . . . 5  |-  ( f  e.  ( ~P G  ^m  om )  ->  f : om --> ~P G )
2 isf32lem11 8739 . . . . . . . . . 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 435 . . . . . . . . 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 1196 . . . . . . . 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 440 . . . . . . 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 124 . . . . . 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 604 . . . . 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 85 . . 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 2880 . 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 8705 . 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 234 1  |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814    C_ wss 3476   ~Pcpw 4010   |^|cint 4282   class class class wbr 4447   suc csuc 4880   ran crn 5000   -->wf 5582   ` cfv 5586  (class class class)co 6282   omcom 6678    ^m cmap 7417    ~<_* cwdom 7979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-1o 7127  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316
This theorem is referenced by:  isf33lem  8742  isfin3-2  8743
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