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Theorem ercpbllem 13775
Description: Lemma for ercpbl 13776. (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 13774 . . 3  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
51, 2, 3divsfval 13774 . . 3  |-  ( ph  ->  ( F `  B
)  =  [ B ]  .~  )
64, 5eqeq12d 2452 . 2  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <->  [ A ]  .~  =  [ B ]  .~  )
)
7 ercpbllem.1 . . 3  |-  ( ph  ->  A  e.  V )
81, 7erth 6951 . 2  |-  ( ph  ->  ( A  .~  B  <->  [ A ]  .~  =  [ B ]  .~  )
)
96, 8bitr4d 249 1  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <-> 
A  .~  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4214    e. cmpt 4268   ` cfv 5456    Er wer 6904   [cec 6905
This theorem is referenced by:  ercpbl  13776  erlecpbl  13777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-er 6907  df-ec 6909
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