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Theorem erlecpbl 13468
Description: Translate the relation compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
erlecpbl.e  |-  ( ph  ->  ( ( A  .~  C  /\  B  .~  D
)  ->  ( A N B  <->  C N D ) ) )
Assertion
Ref Expression
erlecpbl  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( A N B  <-> 
C N D ) ) )
Distinct variable groups:    x,  .~    x, A    x, B    x, C    x, D    x, V    ph, x
Allowed substitution hints:    F( x)    N( x)

Proof of Theorem erlecpbl
StepHypRef Expression
1 ercpbl.r . . . . 5  |-  ( ph  ->  .~  Er  V )
213ad2ant1 976 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  .~  Er  V
)
3 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
433ad2ant1 976 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  V  e.  _V )
5 ercpbl.f . . . 4  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
6 simp2l 981 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
72, 4, 5, 6ercpbllem 13466 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  .~  C ) )
8 simp2r 982 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
92, 4, 5, 8ercpbllem 13466 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  .~  D ) )
107, 9anbi12d 691 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  <->  ( A  .~  C  /\  B  .~  D ) ) )
11 erlecpbl.e . . 3  |-  ( ph  ->  ( ( A  .~  C  /\  B  .~  D
)  ->  ( A N B  <->  C N D ) ) )
12113ad2ant1 976 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( A  .~  C  /\  B  .~  D )  ->  ( A N B  <->  C N D ) ) )
1310, 12sylbid 206 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( A N B  <-> 
C N D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271    Er wer 6673   [cec 6674
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-mo 2161  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-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-er 6676  df-ec 6678
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