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Theorem dff12 5199
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 5007 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
2 funcnv2 5060 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
32anbi2i 445 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
41, 3bitri 182 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287  ∃*wmo 1949   class class class wbr 3837  ccnv 4427  Fun wfun 4996  wf 4998  1-1wf1 4999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-fun 5004  df-f1 5007
This theorem is referenced by:  dff13  5529
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