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Theorem funcnvsn 5163
 Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5166 via cnvsn 5016, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)
Assertion
Ref Expression
funcnvsn

Proof of Theorem funcnvsn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4912 . 2
2 moeq 2854 . . . 4
3 vex 2684 . . . . . . . 8
4 vex 2684 . . . . . . . 8
53, 4brcnv 4717 . . . . . . 7
6 df-br 3925 . . . . . . 7
75, 6bitri 183 . . . . . 6
8 elsni 3540 . . . . . . 7
94, 3opth1 4153 . . . . . . 7
108, 9syl 14 . . . . . 6
117, 10sylbi 120 . . . . 5
1211moimi 2062 . . . 4
132, 12ax-mp 5 . . 3
1413ax-gen 1425 . 2
15 dffun6 5132 . 2
161, 14, 15mpbir2an 926 1
 Colors of variables: wff set class Syntax hints:  wal 1329   wceq 1331   wcel 1480  wmo 1998  csn 3522  cop 3525   class class class wbr 3924  ccnv 4533   wrel 4539   wfun 5112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120 This theorem is referenced by:  funsng  5164
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