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Theorem fliftcnv 6116
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 2454 . . . . 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 6113 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  <. B ,  A >. )  C_  ( S  X.  R ) )
5 relxp 5058 . . . 4  |-  Rel  ( S  X.  R )
6 relss 5038 . . . 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 18 . . 3  |-  ( ph  ->  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )
8 relcnv 5317 . . 3  |-  Rel  `' F
97, 8jctil 537 . 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 6114 . . . . . 6  |-  ( ph  ->  ( z F y  <->  E. x  e.  X  ( z  =  A  /\  y  =  B ) ) )
12 vex 3081 . . . . . . 7  |-  y  e. 
_V
13 vex 3081 . . . . . . 7  |-  z  e. 
_V
1412, 13brcnv 5133 . . . . . 6  |-  ( y `' F z  <->  z F
y )
15 ancom 450 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  A )  <->  ( z  =  A  /\  y  =  B )
)
1615rexbii 2862 . . . . . 6  |-  ( E. x  e.  X  ( y  =  B  /\  z  =  A )  <->  E. x  e.  X  ( z  =  A  /\  y  =  B )
)
1711, 14, 163bitr4g 288 . . . . 5  |-  ( ph  ->  ( y `' F
z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
181, 2, 3fliftel 6114 . . . . 5  |-  ( ph  ->  ( y ran  (
x  e.  X  |->  <. B ,  A >. ) z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
1917, 18bitr4d 256 . . . 4  |-  ( ph  ->  ( y `' F
z  <->  y ran  (
x  e.  X  |->  <. B ,  A >. ) z ) )
20 df-br 4404 . . . 4  |-  ( y `' F z  <->  <. y ,  z >.  e.  `' F )
21 df-br 4404 . . . 4  |-  ( y ran  ( x  e.  X  |->  <. B ,  A >. ) z  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
2219, 20, 213bitr3g 287 . . 3  |-  ( ph  ->  ( <. y ,  z
>.  e.  `' F  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) ) )
2322eqrelrdv2 5050 . 2  |-  ( ( ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )  /\  ph )  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
249, 23mpancom 669 1  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    C_ wss 3439   <.cop 3994   class class class wbr 4403    |-> cmpt 4461    X. cxp 4949   `'ccnv 4950   ran crn 4952   Rel wrel 4956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537
This theorem is referenced by:  pi1xfrcnvlem  20763
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