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Theorem ercpbllem 13450
Description: Lemma for ercpbl 13451. (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 13449 . . 3  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
51, 2, 3divsfval 13449 . . 3  |-  ( ph  ->  ( F `  B
)  =  [ B ]  .~  )
64, 5eqeq12d 2297 . 2  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <->  [ A ]  .~  =  [ B ]  .~  )
)
7 ercpbllem.1 . . 3  |-  ( ph  ->  A  e.  V )
81, 7erth 6704 . 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 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255    Er wer 6657   [cec 6658
This theorem is referenced by:  ercpbl  13451  erlecpbl  13452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-er 6660  df-ec 6662
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