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Theorem fliftcnv 5381
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 2040 . . . . 5  ran  X  |->  <. ,  >.  ran  X  |->  <. ,  >.
2 flift.3 . . . . 5  X  S
3 flift.2 . . . . 5  X  R
41, 2, 3fliftrel 5378 . . . 4  ran  X  |->  <. ,  >.  C_  S  X.  R
5 relxp 4393 . . . 4  Rel  S  X.  R
6 relss 4373 . . . 4  ran  X  |-> 
<. ,  >. 
C_  S  X.  R  Rel  S  X.  R  Rel  ran  X  |->  <. ,  >.
74, 5, 6mpisyl 1335 . . 3  Rel  ran  X  |->  <. ,  >.
8 relcnv 4649 . . 3  Rel  `' F
97, 8jctil 295 . 2  Rel  `' F  Rel  ran  X  |->  <. ,  >.
10 flift.1 . . . . . . 7  F 
ran  X  |->  <. ,  >.
1110, 3, 2fliftel 5379 . . . . . 6  F  X
12 vex 2557 . . . . . . 7 
_V
13 vex 2557 . . . . . . 7 
_V
1412, 13brcnv 4464 . . . . . 6  `' F  F
15 ancom 253 . . . . . . 7
1615rexbii 2328 . . . . . 6  X  X
1711, 14, 163bitr4g 212 . . . . 5  `' F  X
181, 2, 3fliftel 5379 . . . . 5  ran  X  |->  <. ,  >.  X
1917, 18bitr4d 180 . . . 4  `' F  ran  X  |->  <. ,  >.
20 df-br 3759 . . . 4  `' F  <. ,  >.  `' F
21 df-br 3759 . . . 4  ran  X  |->  <. ,  >.  <. ,  >.  ran  X  |-> 
<. ,  >.
2219, 20, 213bitr3g 211 . . 3  <. , 
>.  `' F  <. ,  >.  ran  X  |-> 
<. ,  >.
2322eqrelrdv2 4385 . 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 1243   wcel 1393  wrex 2304    C_ wss 2914   <.cop 3373   class class class wbr 3758    |-> cmpt 3812    X. cxp 4289   `'ccnv 4290   ran crn 4292   Rel wrel 4296
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-iota 4813  df-fun 4850  df-fn 4851  df-f 4852  df-fv 4856
This theorem is referenced by: (None)
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