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Theorem fliftcnv 6000
Description: Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftcnv  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftcnv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2412 . . . . 5  |-  ran  (
x  e.  X  |->  <. B ,  A >. )  =  ran  ( x  e.  X  |->  <. B ,  A >. )
2 flift.3 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
3 flift.2 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
41, 2, 3fliftrel 5997 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  <. B ,  A >. )  C_  ( S  X.  R ) )
5 relxp 4950 . . . 4  |-  Rel  ( S  X.  R )
6 relss 4930 . . . 4  |-  ( ran  ( x  e.  X  |-> 
<. B ,  A >. ) 
C_  ( S  X.  R )  ->  ( Rel  ( S  X.  R
)  ->  Rel  ran  (
x  e.  X  |->  <. B ,  A >. ) ) )
74, 5, 6ee10 1382 . . 3  |-  ( ph  ->  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )
8 relcnv 5209 . . 3  |-  Rel  `' F
97, 8jctil 524 . 2  |-  ( ph  ->  ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) ) )
10 flift.1 . . . . . . 7  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
1110, 3, 2fliftel 5998 . . . . . 6  |-  ( ph  ->  ( z F y  <->  E. x  e.  X  ( z  =  A  /\  y  =  B ) ) )
12 vex 2927 . . . . . . 7  |-  y  e. 
_V
13 vex 2927 . . . . . . 7  |-  z  e. 
_V
1412, 13brcnv 5022 . . . . . 6  |-  ( y `' F z  <->  z F
y )
15 ancom 438 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  A )  <->  ( z  =  A  /\  y  =  B )
)
1615rexbii 2699 . . . . . 6  |-  ( E. x  e.  X  ( y  =  B  /\  z  =  A )  <->  E. x  e.  X  ( z  =  A  /\  y  =  B )
)
1711, 14, 163bitr4g 280 . . . . 5  |-  ( ph  ->  ( y `' F
z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
181, 2, 3fliftel 5998 . . . . 5  |-  ( ph  ->  ( y ran  (
x  e.  X  |->  <. B ,  A >. ) z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
1917, 18bitr4d 248 . . . 4  |-  ( ph  ->  ( y `' F
z  <->  y ran  (
x  e.  X  |->  <. B ,  A >. ) z ) )
20 df-br 4181 . . . 4  |-  ( y `' F z  <->  <. y ,  z >.  e.  `' F )
21 df-br 4181 . . . 4  |-  ( y ran  ( x  e.  X  |->  <. B ,  A >. ) z  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
2219, 20, 213bitr3g 279 . . 3  |-  ( ph  ->  ( <. y ,  z
>.  e.  `' F  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) ) )
2322eqrelrdv2 4942 . 2  |-  ( ( ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )  /\  ph )  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
249, 23mpancom 651 1  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675    C_ wss 3288   <.cop 3785   class class class wbr 4180    e. cmpt 4234    X. cxp 4843   `'ccnv 4844   ran crn 4846   Rel wrel 4850
This theorem is referenced by:  pi1xfrcnvlem  19042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429
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