Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnvsn Structured version   Visualization version   Unicode version

Theorem funcnvsn 5630
 Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5633 via cnvsn 5322, 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 5210 . 2
2 moeq 3216 . . . 4
3 vex 3050 . . . . . . . 8
4 vex 3050 . . . . . . . 8
53, 4brcnv 5020 . . . . . . 7
6 df-br 4406 . . . . . . 7
75, 6bitri 253 . . . . . 6
8 elsni 3995 . . . . . . 7
94, 3opth1 4678 . . . . . . 7
108, 9syl 17 . . . . . 6
117, 10sylbi 199 . . . . 5
1211moimi 2351 . . . 4
132, 12ax-mp 5 . . 3
1413ax-gen 1671 . 2
15 dffun6 5600 . 2
161, 14, 15mpbir2an 932 1
 Colors of variables: wff setvar class Syntax hints:  wal 1444   wceq 1446   wcel 1889  wmo 2302  csn 3970  cop 3976   class class class wbr 4405  ccnv 4836   wrel 4842   wfun 5579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-fun 5587 This theorem is referenced by:  funsng  5631  funcnvpr  5642  funcnvtp  5643  funcnvs1  13010  strlemor1  15229  0spth  25313  2pthlem1  25337  0spth-av  39730  2pthdlem1  39755
 Copyright terms: Public domain W3C validator