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Theorem dff12 6564
 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 6348 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
2 funcnv2 6410 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
32anbi2i 625 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
41, 3bitri 278 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃*wmo 2622   class class class wbr 5052  ◡ccnv 5541  Fun wfun 6337  ⟶wf 6339  –1-1→wf1 6340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-fun 6345  df-f1 6348 This theorem is referenced by:  dff13  7005  fseqenlem2  9449  s4f1o  14280  2ndcdisj  22064  usgrexmplef  27052  phpreu  34986
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