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Theorem itg2lr 22262
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 2457 . . 3  |-  ( S.1 `  G )  =  ( S.1 `  G )
2 breq1 4459 . . . . 5  |-  ( g  =  G  ->  (
g  oR  <_  F 
<->  G  oR  <_  F ) )
3 fveq2 5872 . . . . . 6  |-  ( g  =  G  ->  ( S.1 `  g )  =  ( S.1 `  G
) )
43eqeq2d 2471 . . . . 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 3210 . . 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 22261 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808   class class class wbr 4456   dom cdm 5008   ` cfv 5594    oRcofr 6538    <_ cle 9646   S.1citg1 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602
This theorem is referenced by:  itg2ub  22265
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