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Theorem fliftcnv 5466
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 2082 . . . . 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 5463 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  <. B ,  A >. )  C_  ( S  X.  R ) )
5 relxp 4475 . . . 4  |-  Rel  ( S  X.  R )
6 relss 4453 . . . 4  |-  ( ran  ( x  e.  X  |-> 
<. B ,  A >. ) 
C_  ( S  X.  R )  ->  ( Rel  ( S  X.  R
)  ->  Rel  ran  (
x  e.  X  |->  <. B ,  A >. ) ) )
74, 5, 6mpisyl 1376 . . 3  |-  ( ph  ->  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )
8 relcnv 4733 . . 3  |-  Rel  `' F
97, 8jctil 305 . 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 5464 . . . . . 6  |-  ( ph  ->  ( z F y  <->  E. x  e.  X  ( z  =  A  /\  y  =  B ) ) )
12 vex 2605 . . . . . . 7  |-  y  e. 
_V
13 vex 2605 . . . . . . 7  |-  z  e. 
_V
1412, 13brcnv 4546 . . . . . 6  |-  ( y `' F z  <->  z F
y )
15 ancom 262 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  A )  <->  ( z  =  A  /\  y  =  B )
)
1615rexbii 2374 . . . . . 6  |-  ( E. x  e.  X  ( y  =  B  /\  z  =  A )  <->  E. x  e.  X  ( z  =  A  /\  y  =  B )
)
1711, 14, 163bitr4g 221 . . . . 5  |-  ( ph  ->  ( y `' F
z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
181, 2, 3fliftel 5464 . . . . 5  |-  ( ph  ->  ( y ran  (
x  e.  X  |->  <. B ,  A >. ) z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
1917, 18bitr4d 189 . . . 4  |-  ( ph  ->  ( y `' F
z  <->  y ran  (
x  e.  X  |->  <. B ,  A >. ) z ) )
20 df-br 3794 . . . 4  |-  ( y `' F z  <->  <. y ,  z >.  e.  `' F )
21 df-br 3794 . . . 4  |-  ( y ran  ( x  e.  X  |->  <. B ,  A >. ) z  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
2219, 20, 213bitr3g 220 . . 3  |-  ( ph  ->  ( <. y ,  z
>.  e.  `' F  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) ) )
2322eqrelrdv2 4465 . 2  |-  ( ( ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )  /\  ph )  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
249, 23mpancom 413 1  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   E.wrex 2350    C_ wss 2974   <.cop 3409   class class class wbr 3793    |-> cmpt 3847    X. cxp 4369   `'ccnv 4370   ran crn 4372   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940
This theorem is referenced by: (None)
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