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Theorem elcnvlem 36278
 Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
Hypothesis
Ref Expression
elcnvlem.f
Assertion
Ref Expression
elcnvlem

Proof of Theorem elcnvlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnv2 5017 . 2
2 fveq2 5879 . . . . 5
3 vex 3034 . . . . . . 7
4 vex 3034 . . . . . . 7
53, 4opelvv 4886 . . . . . 6
63, 4op2ndd 6823 . . . . . . . 8
73, 4op1std 6822 . . . . . . . 8
86, 7opeq12d 4166 . . . . . . 7
9 elcnvlem.f . . . . . . 7
10 opex 4664 . . . . . . 7
118, 9, 10fvmpt 5963 . . . . . 6
125, 11ax-mp 5 . . . . 5
132, 12syl6eq 2521 . . . 4
1413eleq1d 2533 . . 3
1514copsex2gb 4950 . 2
161, 15bitri 257 1
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  cvv 3031  cop 3965   cmpt 4454   cxp 4837  ccnv 4838  cfv 5589  c1st 6810  c2nd 6811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813 This theorem is referenced by:  elcnvintab  36279
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