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Theorem inf3lema 7505
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7516 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 3471 . . 3  |-  ( f  =  A  ->  (
f  i^i  x )  =  ( A  i^i  x ) )
21sseq1d 3311 . 2  |-  ( f  =  A  ->  (
( f  i^i  x
)  C_  B  <->  ( A  i^i  x )  C_  B
) )
3 inf3lem.4 . . 3  |-  B  e. 
_V
4 sseq2 3306 . . . . 5  |-  ( v  =  B  ->  (
( f  i^i  x
)  C_  v  <->  ( f  i^i  x )  C_  B
) )
54rabbidv 2884 . . . 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 3306 . . . . . . . 8  |-  ( y  =  v  ->  (
( w  i^i  x
)  C_  y  <->  ( w  i^i  x )  C_  v
) )
87rabbidv 2884 . . . . . . 7  |-  ( y  =  v  ->  { w  e.  x  |  (
w  i^i  x )  C_  y }  =  {
w  e.  x  |  ( w  i^i  x
)  C_  v }
)
9 ineq1 3471 . . . . . . . . 9  |-  ( w  =  f  ->  (
w  i^i  x )  =  ( f  i^i  x ) )
109sseq1d 3311 . . . . . . . 8  |-  ( w  =  f  ->  (
( w  i^i  x
)  C_  v  <->  ( f  i^i  x )  C_  v
) )
1110cbvrabv 2891 . . . . . . 7  |-  { w  e.  x  |  (
w  i^i  x )  C_  v }  =  {
f  e.  x  |  ( f  i^i  x
)  C_  v }
128, 11syl6eq 2428 . . . . . 6  |-  ( y  =  v  ->  { w  e.  x  |  (
w  i^i  x )  C_  y }  =  {
f  e.  x  |  ( f  i^i  x
)  C_  v }
)
1312cbvmptv 4234 . . . . 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 2400 . . . 4  |-  G  =  ( v  e.  _V  |->  { f  e.  x  |  ( f  i^i  x )  C_  v } )
15 vex 2895 . . . . 5  |-  x  e. 
_V
1615rabex 4288 . . . 4  |-  { f  e.  x  |  ( f  i^i  x ) 
C_  B }  e.  _V
175, 14, 16fvmpt 5738 . . 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 3030 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 1649    e. wcel 1717   {crab 2646   _Vcvv 2892    i^i cin 3255    C_ wss 3256   (/)c0 3564    e. cmpt 4200   omcom 4778    |` cres 4813   ` cfv 5387   reccrdg 6596
This theorem is referenced by:  inf3lemd  7508  inf3lem1  7509  inf3lem2  7510  inf3lem3  7511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395
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