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Theorem fliftcnv 5894
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 2358 . . . . 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 5891 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  <. B ,  A >. )  C_  ( S  X.  R ) )
5 relxp 4873 . . . 4  |-  Rel  ( S  X.  R )
6 relss 4854 . . . 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 1376 . . 3  |-  ( ph  ->  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )
8 relcnv 5130 . . 3  |-  Rel  `' F
97, 8jctil 523 . 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 5892 . . . . . 6  |-  ( ph  ->  ( z F y  <->  E. x  e.  X  ( z  =  A  /\  y  =  B ) ) )
12 vex 2867 . . . . . . 7  |-  y  e. 
_V
13 vex 2867 . . . . . . 7  |-  z  e. 
_V
1412, 13brcnv 4943 . . . . . 6  |-  ( y `' F z  <->  z F
y )
15 ancom 437 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  A )  <->  ( z  =  A  /\  y  =  B )
)
1615rexbii 2644 . . . . . 6  |-  ( E. x  e.  X  ( y  =  B  /\  z  =  A )  <->  E. x  e.  X  ( z  =  A  /\  y  =  B )
)
1711, 14, 163bitr4g 279 . . . . 5  |-  ( ph  ->  ( y `' F
z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
181, 2, 3fliftel 5892 . . . . 5  |-  ( ph  ->  ( y ran  (
x  e.  X  |->  <. B ,  A >. ) z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
1917, 18bitr4d 247 . . . 4  |-  ( ph  ->  ( y `' F
z  <->  y ran  (
x  e.  X  |->  <. B ,  A >. ) z ) )
20 df-br 4103 . . . 4  |-  ( y `' F z  <->  <. y ,  z >.  e.  `' F )
21 df-br 4103 . . . 4  |-  ( y ran  ( x  e.  X  |->  <. B ,  A >. ) z  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
2219, 20, 213bitr3g 278 . . 3  |-  ( ph  ->  ( <. y ,  z
>.  e.  `' F  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) ) )
2322eqrelrdv2 4865 . 2  |-  ( ( ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )  /\  ph )  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
249, 23mpancom 650 1  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620    C_ wss 3228   <.cop 3719   class class class wbr 4102    e. cmpt 4156    X. cxp 4766   `'ccnv 4767   ran crn 4769   Rel wrel 4773
This theorem is referenced by:  pi1xfrcnvlem  18652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342
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