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Theorem itg2lr 19614
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  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2lr  |-  ( ( G  e.  dom  S.1  /\  G  o R  <_  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 2435 . . 3  |-  ( S.1 `  G )  =  ( S.1 `  G )
2 breq1 4207 . . . . 5  |-  ( g  =  G  ->  (
g  o R  <_  F 
<->  G  o R  <_  F ) )
3 fveq2 5720 . . . . . 6  |-  ( g  =  G  ->  ( S.1 `  g )  =  ( S.1 `  G
) )
43eqeq2d 2446 . . . . 5  |-  ( g  =  G  ->  (
( S.1 `  G )  =  ( S.1 `  g
)  <->  ( S.1 `  G
)  =  ( S.1 `  G ) ) )
52, 4anbi12d 692 . . . 4  |-  ( g  =  G  ->  (
( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) )  <->  ( G  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  G
) ) ) )
65rspcev 3044 . . 3  |-  ( ( G  e.  dom  S.1  /\  ( G  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  G
) ) )  ->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
71, 6mpanr2 666 . 2  |-  ( ( G  e.  dom  S.1  /\  G  o R  <_  F )  ->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
8 itg2val.1 . . 3  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
98itg2l 19613 . 2  |-  ( ( S.1 `  G )  e.  L  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
107, 9sylibr 204 1  |-  ( ( G  e.  dom  S.1  /\  G  o R  <_  F )  ->  ( S.1 `  G )  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   class class class wbr 4204   dom cdm 4870   ` cfv 5446    o Rcofr 6296    <_ cle 9113   S.1citg1 19499
This theorem is referenced by:  itg2ub  19617
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
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-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-iota 5410  df-fv 5454
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