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Theorem cnvin 4731
 Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin

Proof of Theorem cnvin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4353 . . 3
2 inopab 4468 . . . 4
3 brin 3811 . . . . 5
43opabbii 3824 . . . 4
52, 4eqtr4i 2063 . . 3
61, 5eqtr4i 2063 . 2
7 df-cnv 4353 . . 3
8 df-cnv 4353 . . 3
97, 8ineq12i 3136 . 2
106, 9eqtr4i 2063 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243   cin 2916   class class class wbr 3764  copab 3817  ccnv 4344 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 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353 This theorem is referenced by:  rnin  4733  dminxp  4765  imainrect  4766  cnvcnv  4773
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