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Theorem fliftcnv 5378
Description: Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  F 
ran  X  |->  <. ,  >.
flift.2  X  R
flift.3  X  S
Assertion
Ref Expression
fliftcnv  `' F  ran  X  |-> 
<. ,  >.
Distinct variable groups:   , R   ,   , X   , S
Allowed substitution hints:   ()   ()    F()

Proof of Theorem fliftcnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . . . 5  ran  X  |->  <. ,  >.  ran  X  |->  <. ,  >.
2 flift.3 . . . . 5  X  S
3 flift.2 . . . . 5  X  R
41, 2, 3fliftrel 5375 . . . 4  ran  X  |->  <. ,  >.  C_  S  X.  R
5 relxp 4390 . . . 4  Rel  S  X.  R
6 relss 4370 . . . 4  ran  X  |-> 
<. ,  >. 
C_  S  X.  R  Rel  S  X.  R  Rel  ran  X  |->  <. ,  >.
74, 5, 6mpisyl 1332 . . 3  Rel  ran  X  |->  <. ,  >.
8 relcnv 4646 . . 3  Rel  `' F
97, 8jctil 295 . 2  Rel  `' F  Rel  ran  X  |->  <. ,  >.
10 flift.1 . . . . . . 7  F 
ran  X  |->  <. ,  >.
1110, 3, 2fliftel 5376 . . . . . 6  F  X
12 vex 2554 . . . . . . 7 
_V
13 vex 2554 . . . . . . 7 
_V
1412, 13brcnv 4461 . . . . . 6  `' F  F
15 ancom 253 . . . . . . 7
1615rexbii 2325 . . . . . 6  X  X
1711, 14, 163bitr4g 212 . . . . 5  `' F  X
181, 2, 3fliftel 5376 . . . . 5  ran  X  |->  <. ,  >.  X
1917, 18bitr4d 180 . . . 4  `' F  ran  X  |->  <. ,  >.
20 df-br 3756 . . . 4  `' F  <. ,  >.  `' F
21 df-br 3756 . . . 4  ran  X  |->  <. ,  >.  <. ,  >.  ran  X  |-> 
<. ,  >.
2219, 20, 213bitr3g 211 . . 3  <. , 
>.  `' F  <. ,  >.  ran  X  |-> 
<. ,  >.
2322eqrelrdv2 4382 . 2  Rel  `' F  Rel  ran  X  |->  <. ,  >.  `' F  ran  X  |->  <. ,  >.
249, 23mpancom 399 1  `' F  ran  X  |-> 
<. ,  >.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wrex 2301    C_ wss 2911   <.cop 3370   class class class wbr 3755    |-> cmpt 3809    X. cxp 4286   `'ccnv 4287   ran crn 4289   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by: (None)
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