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Theorem inf3lema 7571
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7582 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
inf3lem.2  |-  F  =  ( rec ( G ,  (/) )  |`  om )
inf3lem.3  |-  A  e. 
_V
inf3lem.4  |-  B  e. 
_V
Assertion
Ref Expression
inf3lema  |-  ( A  e.  ( G `  B )  <->  ( A  e.  x  /\  ( A  i^i  x )  C_  B ) )
Distinct variable group:    x, y, w
Allowed substitution hints:    A( x, y, w)    B( x, y, w)    F( x, y, w)    G( x, y, w)

Proof of Theorem inf3lema
Dummy variables  v 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 3527 . . 3  |-  ( f  =  A  ->  (
f  i^i  x )  =  ( A  i^i  x ) )
21sseq1d 3367 . 2  |-  ( f  =  A  ->  (
( f  i^i  x
)  C_  B  <->  ( A  i^i  x )  C_  B
) )
3 inf3lem.4 . . 3  |-  B  e. 
_V
4 sseq2 3362 . . . . 5  |-  ( v  =  B  ->  (
( f  i^i  x
)  C_  v  <->  ( f  i^i  x )  C_  B
) )
54rabbidv 2940 . . . 4  |-  ( v  =  B  ->  { f  e.  x  |  ( f  i^i  x ) 
C_  v }  =  { f  e.  x  |  ( f  i^i  x )  C_  B } )
6 inf3lem.1 . . . . 5  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
7 sseq2 3362 . . . . . . . 8  |-  ( y  =  v  ->  (
( w  i^i  x
)  C_  y  <->  ( w  i^i  x )  C_  v
) )
87rabbidv 2940 . . . . . . 7  |-  ( y  =  v  ->  { w  e.  x  |  (
w  i^i  x )  C_  y }  =  {
w  e.  x  |  ( w  i^i  x
)  C_  v }
)
9 ineq1 3527 . . . . . . . . 9  |-  ( w  =  f  ->  (
w  i^i  x )  =  ( f  i^i  x ) )
109sseq1d 3367 . . . . . . . 8  |-  ( w  =  f  ->  (
( w  i^i  x
)  C_  v  <->  ( f  i^i  x )  C_  v
) )
1110cbvrabv 2947 . . . . . . 7  |-  { w  e.  x  |  (
w  i^i  x )  C_  v }  =  {
f  e.  x  |  ( f  i^i  x
)  C_  v }
128, 11syl6eq 2483 . . . . . 6  |-  ( y  =  v  ->  { w  e.  x  |  (
w  i^i  x )  C_  y }  =  {
f  e.  x  |  ( f  i^i  x
)  C_  v }
)
1312cbvmptv 4292 . . . . 5  |-  ( y  e.  _V  |->  { w  e.  x  |  (
w  i^i  x )  C_  y } )  =  ( v  e.  _V  |->  { f  e.  x  |  ( f  i^i  x )  C_  v } )
146, 13eqtri 2455 . . . 4  |-  G  =  ( v  e.  _V  |->  { f  e.  x  |  ( f  i^i  x )  C_  v } )
15 vex 2951 . . . . 5  |-  x  e. 
_V
1615rabex 4346 . . . 4  |-  { f  e.  x  |  ( f  i^i  x ) 
C_  B }  e.  _V
175, 14, 16fvmpt 5798 . . 3  |-  ( B  e.  _V  ->  ( G `  B )  =  { f  e.  x  |  ( f  i^i  x )  C_  B } )
183, 17ax-mp 8 . 2  |-  ( G `
 B )  =  { f  e.  x  |  ( f  i^i  x )  C_  B }
192, 18elrab2 3086 1  |-  ( A  e.  ( G `  B )  <->  ( A  e.  x  /\  ( A  i^i  x )  C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620    e. cmpt 4258   omcom 4837    |` cres 4872   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  inf3lemd  7574  inf3lem1  7575  inf3lem2  7576  inf3lem3  7577
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454
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