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Theorem dff1o2 6283
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 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 6038 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 df-f1 6036 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
3 df-fo 6037 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 604 . . 3 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 anass 459 . . . 4 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6 3anan12 1081 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
76anbi1i 602 . . . . 5 (((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
8 eqimss 3806 . . . . . . . 8 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
9 df-f 6035 . . . . . . . . 9 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
109biimpri 218 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → 𝐹:𝐴𝐵)
118, 10sylan2 572 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
12113adant2 1125 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
1312pm4.71i 541 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵))
14 ancom 452 . . . . 5 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
157, 13, 143bitr4ri 293 . . . 4 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
165, 15bitri 264 . . 3 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
174, 16bitri 264 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
181, 17bitri 264 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wss 3723  ccnv 5248  ran crn 5250  Fun wfun 6025   Fn wfn 6026  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-in 3730  df-ss 3737  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  dff1o3  6284  dff1o4  6286  f1orn  6288  tz7.49c  7693  fiint  8392  symgfixelsi  18061  dfrelog  24532  adj1o  29090  fresf1o  29770  f1mptrn  29772  esumc  30450  ntrneinex  38897  stoweidlem39  40769
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