MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftcnv Structured version   Unicode version

Theorem fliftcnv 6069
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 2443 . . . . 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 6066 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  <. B ,  A >. )  C_  ( S  X.  R ) )
5 relxp 5018 . . . 4  |-  Rel  ( S  X.  R )
6 relss 4998 . . . 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 1386 . . 3  |-  ( ph  ->  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )
8 relcnv 5277 . . 3  |-  Rel  `' F
97, 8jctil 525 . 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 6067 . . . . . 6  |-  ( ph  ->  ( z F y  <->  E. x  e.  X  ( z  =  A  /\  y  =  B ) ) )
12 vex 2968 . . . . . . 7  |-  y  e. 
_V
13 vex 2968 . . . . . . 7  |-  z  e. 
_V
1412, 13brcnv 5090 . . . . . 6  |-  ( y `' F z  <->  z F
y )
15 ancom 439 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  A )  <->  ( z  =  A  /\  y  =  B )
)
1615rexbii 2737 . . . . . 6  |-  ( E. x  e.  X  ( y  =  B  /\  z  =  A )  <->  E. x  e.  X  ( z  =  A  /\  y  =  B )
)
1711, 14, 163bitr4g 281 . . . . 5  |-  ( ph  ->  ( y `' F
z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
181, 2, 3fliftel 6067 . . . . 5  |-  ( ph  ->  ( y ran  (
x  e.  X  |->  <. B ,  A >. ) z  <->  E. x  e.  X  ( y  =  B  /\  z  =  A ) ) )
1917, 18bitr4d 249 . . . 4  |-  ( ph  ->  ( y `' F
z  <->  y ran  (
x  e.  X  |->  <. B ,  A >. ) z ) )
20 df-br 4244 . . . 4  |-  ( y `' F z  <->  <. y ,  z >.  e.  `' F )
21 df-br 4244 . . . 4  |-  ( y ran  ( x  e.  X  |->  <. B ,  A >. ) z  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
2219, 20, 213bitr3g 280 . . 3  |-  ( ph  ->  ( <. y ,  z
>.  e.  `' F  <->  <. y ,  z >.  e.  ran  ( x  e.  X  |-> 
<. B ,  A >. ) ) )
2322eqrelrdv2 5010 . 2  |-  ( ( ( Rel  `' F  /\  Rel  ran  ( x  e.  X  |->  <. B ,  A >. ) )  /\  ph )  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
249, 23mpancom 652 1  |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |-> 
<. B ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   E.wrex 2713    C_ wss 3309   <.cop 3846   class class class wbr 4243    e. cmpt 4297    X. cxp 4911   `'ccnv 4912   ran crn 4914   Rel wrel 4918
This theorem is referenced by:  pi1xfrcnvlem  19119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fv 5497
  Copyright terms: Public domain W3C validator