MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff12 Structured version   Visualization version   GIF version

Theorem dff12 6738
Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
Assertion
Ref Expression
dff12 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
Distinct variable group:   𝑥,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem dff12
StepHypRef Expression
1 df-f1 6502 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
2 funcnv2 6570 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
32anbi2i 624 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
41, 3bitri 275 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wal 1540  ∃*wmo 2537   class class class wbr 5106  ccnv 5633  Fun wfun 6491  wf 6493  1-1wf1 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-fun 6499  df-f1 6502
This theorem is referenced by:  dff13  7203  fseqenlem2  9962  s4f1o  14808  2ndcdisj  22810  usgrexmplef  28210  phpreu  36065
  Copyright terms: Public domain W3C validator