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Theorem fliftcnv 6209
Description: Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1
flift.2
flift.3
Assertion
Ref Expression
fliftcnv
Distinct variable groups:   ,   ,   ,   ,S

Proof of Theorem fliftcnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . . 5
2 flift.3 . . . . 5
3 flift.2 . . . . 5
41, 2, 3fliftrel 6206 . . . 4
5 relxp 5115 . . . 4
6 relss 5095 . . . 4
74, 5, 6mpisyl 18 . . 3
8 relcnv 5379 . . 3
97, 8jctil 537 . 2
10 flift.1 . . . . . . 7
1110, 3, 2fliftel 6207 . . . . . 6
12 vex 3112 . . . . . . 7
13 vex 3112 . . . . . . 7
1412, 13brcnv 5190 . . . . . 6
15 ancom 450 . . . . . . 7
1615rexbii 2959 . . . . . 6
1711, 14, 163bitr4g 288 . . . . 5
181, 2, 3fliftel 6207 . . . . 5
1917, 18bitr4d 256 . . . 4
20 df-br 4453 . . . 4
21 df-br 4453 . . . 4
2219, 20, 213bitr3g 287 . . 3
2322eqrelrdv2 5107 . 2
249, 23mpancom 669 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808  C_wss 3475  <.cop 4035   class class class wbr 4452  e.cmpt 4510  X.cxp 5002  `'ccnv 5003  rancrn 5005  Relwrel 5009
This theorem is referenced by:  pi1xfrcnvlem  21556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601
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