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Theorem ercpbllem 13499
Description: Lemma for ercpbl 13500. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
ercpbllem.1  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
ercpbllem  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <-> 
A  .~  B )
)
Distinct variable groups:    x,  .~    x, A    x, B    x, V    ph, x
Allowed substitution hint:    F( x)

Proof of Theorem ercpbllem
StepHypRef Expression
1 ercpbl.r . . . 4  |-  ( ph  ->  .~  Er  V )
2 ercpbl.v . . . 4  |-  ( ph  ->  V  e.  _V )
3 ercpbl.f . . . 4  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
41, 2, 3divsfval 13498 . . 3  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
51, 2, 3divsfval 13498 . . 3  |-  ( ph  ->  ( F `  B
)  =  [ B ]  .~  )
64, 5eqeq12d 2330 . 2  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <->  [ A ]  .~  =  [ B ]  .~  )
)
7 ercpbllem.1 . . 3  |-  ( ph  ->  A  e.  V )
81, 7erth 6746 . 2  |-  ( ph  ->  ( A  .~  B  <->  [ A ]  .~  =  [ B ]  .~  )
)
96, 8bitr4d 247 1  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <-> 
A  .~  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   _Vcvv 2822   class class class wbr 4060    e. cmpt 4114   ` cfv 5292    Er wer 6699   [cec 6700
This theorem is referenced by:  ercpbl  13500  erlecpbl  13501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-er 6702  df-ec 6704
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