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Theorem dff1o2 5291
 Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)
Assertion
Ref Expression
dff1o2 (F:A1-1-ontoB ↔ (F Fn A Fun F ran F = B))

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 4794 . 2 (F:A1-1-ontoB ↔ (F:A1-1B F:AontoB))
2 df-f1 4792 . . 3 (F:A1-1B ↔ (F:A–→B Fun F))
3 df-fo 4793 . . 3 (F:AontoB ↔ (F Fn A ran F = B))
42, 3anbi12i 678 . 2 ((F:A1-1B F:AontoB) ↔ ((F:A–→B Fun F) (F Fn A ran F = B)))
5 ancom 437 . . . 4 ((F:A–→B (Fun F (F Fn A ran F = B))) ↔ ((Fun F (F Fn A ran F = B)) F:A–→B))
6 3anass 938 . . . . . 6 ((F Fn A Fun F ran F = B) ↔ (F Fn A (Fun F ran F = B)))
7 an12 772 . . . . . 6 ((F Fn A (Fun F ran F = B)) ↔ (Fun F (F Fn A ran F = B)))
86, 7bitri 240 . . . . 5 ((F Fn A Fun F ran F = B) ↔ (Fun F (F Fn A ran F = B)))
98anbi1i 676 . . . 4 (((F Fn A Fun F ran F = B) F:A–→B) ↔ ((Fun F (F Fn A ran F = B)) F:A–→B))
105, 9bitr4i 243 . . 3 ((F:A–→B (Fun F (F Fn A ran F = B))) ↔ ((F Fn A Fun F ran F = B) F:A–→B))
11 anass 630 . . 3 (((F:A–→B Fun F) (F Fn A ran F = B)) ↔ (F:A–→B (Fun F (F Fn A ran F = B))))
12 eqimss 3323 . . . . . 6 (ran F = B → ran F B)
13 df-f 4791 . . . . . . 7 (F:A–→B ↔ (F Fn A ran F B))
1413biimpri 197 . . . . . 6 ((F Fn A ran F B) → F:A–→B)
1512, 14sylan2 460 . . . . 5 ((F Fn A ran F = B) → F:A–→B)
16153adant2 974 . . . 4 ((F Fn A Fun F ran F = B) → F:A–→B)
1716pm4.71i 613 . . 3 ((F Fn A Fun F ran F = B) ↔ ((F Fn A Fun F ran F = B) F:A–→B))
1810, 11, 173bitr4i 268 . 2 (((F:A–→B Fun F) (F Fn A ran F = B)) ↔ (F Fn A Fun F ran F = B))
191, 4, 183bitri 262 1 (F:A1-1-ontoB ↔ (F Fn A Fun F ran F = B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ⊆ wss 3257  ◡ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  dff1o3  5292  dff1o4  5294  f1orn  5296  fundmen  6043  enmap1  6074  enprmap  6082  sbthlem3  6205
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