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Theorem itg2lr 21213
Description: Sufficient condition for elementhood in the set  L. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2lr  |-  ( ( G  e.  dom  S.1  /\  G  oR  <_  F )  ->  ( S.1 `  G )  e.  L )
Distinct variable groups:    x, g, F    g, G, x
Allowed substitution hints:    L( x, g)

Proof of Theorem itg2lr
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( S.1 `  G )  =  ( S.1 `  G )
2 breq1 4300 . . . . 5  |-  ( g  =  G  ->  (
g  oR  <_  F 
<->  G  oR  <_  F ) )
3 fveq2 5696 . . . . . 6  |-  ( g  =  G  ->  ( S.1 `  g )  =  ( S.1 `  G
) )
43eqeq2d 2454 . . . . 5  |-  ( g  =  G  ->  (
( S.1 `  G )  =  ( S.1 `  g
)  <->  ( S.1 `  G
)  =  ( S.1 `  G ) ) )
52, 4anbi12d 710 . . . 4  |-  ( g  =  G  ->  (
( g  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) )  <->  ( G  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  G
) ) ) )
65rspcev 3078 . . 3  |-  ( ( G  e.  dom  S.1  /\  ( G  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  G
) ) )  ->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
71, 6mpanr2 684 . 2  |-  ( ( G  e.  dom  S.1  /\  G  oR  <_  F )  ->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
8 itg2val.1 . . 3  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
98itg2l 21212 . 2  |-  ( ( S.1 `  G )  e.  L  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
107, 9sylibr 212 1  |-  ( ( G  e.  dom  S.1  /\  G  oR  <_  F )  ->  ( S.1 `  G )  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721   class class class wbr 4297   dom cdm 4845   ` cfv 5423    oRcofr 6324    <_ cle 9424   S.1citg1 21100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431
This theorem is referenced by:  itg2ub  21216
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