ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o4 GIF version

Theorem dff1o4 5341
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Proof of Theorem dff1o4
StepHypRef Expression
1 dff1o2 5338 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
2 3anass 949 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)))
3 df-rn 4518 . . . . . 6 ran 𝐹 = dom 𝐹
43eqeq1i 2123 . . . . 5 (ran 𝐹 = 𝐵 ↔ dom 𝐹 = 𝐵)
54anbi2i 450 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
6 df-fn 5094 . . . 4 (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
75, 6bitr4i 186 . . 3 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ 𝐹 Fn 𝐵)
87anbi2i 450 . 2 ((𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
91, 2, 83bitri 205 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  w3a 945   = wceq 1314  ccnv 4506  dom cdm 4507  ran crn 4508  Fun wfun 5085   Fn wfn 5086  1-1-ontowf1o 5090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052  df-rn 4518  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098
This theorem is referenced by:  f1ocnv  5346  f1oun  5353  f1o00  5368  f1oi  5371  f1osn  5373  f1ompt  5537  f1ofveu  5728  f1ocnvd  5938  f1od2  6098  mapsnf1o2  6556  sbthlemi9  6819  hmeof1o2  12383
  Copyright terms: Public domain W3C validator