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Theorem funcnvsn 5638
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5641 via cnvsn 5496, 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 5379 . 2
2 moeq 3275 . . . 4
3 vex 3112 . . . . . . . 8
4 vex 3112 . . . . . . . 8
53, 4brcnv 5190 . . . . . . 7
6 df-br 4453 . . . . . . 7
75, 6bitri 249 . . . . . 6
8 elsni 4054 . . . . . . 7
94, 3opth1 4725 . . . . . . 7
108, 9syl 16 . . . . . 6
117, 10sylbi 195 . . . . 5
1211moimi 2340 . . . 4
132, 12ax-mp 5 . . 3
1413ax-gen 1618 . 2
15 dffun6 5608 . 2
161, 14, 15mpbir2an 920 1
Colors of variables: wff setvar class
Syntax hints:  A.wal 1393  =wceq 1395  e.wcel 1818  E*wmo 2283  {csn 4029  <.cop 4035   class class class wbr 4452  `'ccnv 5003  Relwrel 5009  Funwfun 5587
This theorem is referenced by:  funsng  5639  strlemor1  14724  0spth  24573  2pthlem1  24597
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-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-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-fun 5595
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