ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funcnvsn Structured version   GIF version

Theorem funcnvsn 4888
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 4891 via cnvsn 4746, 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 4646 . 2 Rel {⟨A, B⟩}
2 moeq 2710 . . . 4 ∃*y y = A
3 vex 2554 . . . . . . . 8 x V
4 vex 2554 . . . . . . . 8 y V
53, 4brcnv 4461 . . . . . . 7 (x{⟨A, B⟩}yy{⟨A, B⟩}x)
6 df-br 3756 . . . . . . 7 (y{⟨A, B⟩}x ↔ ⟨y, x {⟨A, B⟩})
75, 6bitri 173 . . . . . 6 (x{⟨A, B⟩}y ↔ ⟨y, x {⟨A, B⟩})
8 elsni 3391 . . . . . . 7 (⟨y, x {⟨A, B⟩} → ⟨y, x⟩ = ⟨A, B⟩)
94, 3opth1 3964 . . . . . . 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 1962 . . . 4 (∃*y y = A∃*y x{⟨A, B⟩}y)
132, 12ax-mp 7 . . 3 ∃*y x{⟨A, B⟩}y
1413ax-gen 1335 . 2 x∃*y x{⟨A, B⟩}y
15 dffun6 4859 . 2 (Fun {⟨A, B⟩} ↔ (Rel {⟨A, B⟩} x∃*y x{⟨A, B⟩}y))
161, 14, 15mpbir2an 848 1 Fun {⟨A, B⟩}
Colors of variables: wff set class
Syntax hints:  wal 1240   = wceq 1242   wcel 1390  ∃*wmo 1898  {csn 3367  cop 3370   class class class wbr 3755  ccnv 4287  Rel wrel 4293  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by:  funsng  4889
  Copyright terms: Public domain W3C validator