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Theorem itg2lr 19101
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 2296 . . 3  |-  ( S.1 `  G )  =  ( S.1 `  G )
2 breq1 4042 . . . . 5  |-  ( g  =  G  ->  (
g  o R  <_  F 
<->  G  o R  <_  F ) )
3 fveq2 5541 . . . . . 6  |-  ( g  =  G  ->  ( S.1 `  g )  =  ( S.1 `  G
) )
43eqeq2d 2307 . . . . 5  |-  ( g  =  G  ->  (
( S.1 `  G )  =  ( S.1 `  g
)  <->  ( S.1 `  G
)  =  ( S.1 `  G ) ) )
52, 4anbi12d 691 . . . 4  |-  ( g  =  G  ->  (
( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) )  <->  ( G  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  G
) ) ) )
65rspcev 2897 . . 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 665 . 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 19100 . 2  |-  ( ( S.1 `  G )  e.  L  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  ( S.1 `  G )  =  ( S.1 `  g
) ) )
107, 9sylibr 203 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 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   class class class wbr 4039   dom cdm 4705   ` cfv 5271    o Rcofr 6093    <_ cle 8884   S.1citg1 18986
This theorem is referenced by:  itg2ub  19104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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