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Theorem funcnvsn 4867
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 4870 via cnvsn 4726, but stating it this way allows us to skip the sethood assumptions on A and B. (Contributed by NM, 30-Apr-2015.)
Assertion
Ref Expression
funcnvsn Fun {⟨A, B⟩}

Proof of Theorem funcnvsn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4626 . 2 Rel {⟨A, B⟩}
2 moeq 2689 . . . 4 ∃*y y = A
3 vex 2534 . . . . . . . 8 x V
4 vex 2534 . . . . . . . 8 y V
53, 4brcnv 4441 . . . . . . 7 (x{⟨A, B⟩}yy{⟨A, B⟩}x)
6 df-br 3735 . . . . . . 7 (y{⟨A, B⟩}x ↔ ⟨y, x {⟨A, B⟩})
75, 6bitri 173 . . . . . 6 (x{⟨A, B⟩}y ↔ ⟨y, x {⟨A, B⟩})
8 elsni 3370 . . . . . . 7 (⟨y, x {⟨A, B⟩} → ⟨y, x⟩ = ⟨A, B⟩)
94, 3opth1 3943 . . . . . . 7 (⟨y, x⟩ = ⟨A, B⟩ → y = A)
108, 9syl 14 . . . . . 6 (⟨y, x {⟨A, B⟩} → y = A)
117, 10sylbi 114 . . . . 5 (x{⟨A, B⟩}yy = A)
1211moimi 1943 . . . 4 (∃*y y = A∃*y x{⟨A, B⟩}y)
132, 12ax-mp 7 . . 3 ∃*y x{⟨A, B⟩}y
1413ax-gen 1314 . 2 x∃*y x{⟨A, B⟩}y
15 dffun6 4838 . 2 (Fun {⟨A, B⟩} ↔ (Rel {⟨A, B⟩} x∃*y x{⟨A, B⟩}y))
161, 14, 15mpbir2an 835 1 Fun {⟨A, B⟩}
Colors of variables: wff set class
Syntax hints:  wal 1224   = wceq 1226   wcel 1370  ∃*wmo 1879  {csn 3346  cop 3349   class class class wbr 3734  ccnv 4267  Rel wrel 4273  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-fun 4827
This theorem is referenced by:  funsng  4868
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