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Theorem dff1o2 5056
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 4836 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
2 df-f1 4834 . . . 4 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
3 df-fo 4835 . . . 4 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
42, 3anbi12i 436 . . 3 ((𝐹:A1-1B 𝐹:AontoB) ↔ ((𝐹:AB Fun 𝐹) (𝐹 Fn A ran 𝐹 = B)))
5 anass 383 . . . 4 (((𝐹:AB Fun 𝐹) (𝐹 Fn A ran 𝐹 = B)) ↔ (𝐹:AB (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B))))
6 3anan12 885 . . . . . 6 ((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) ↔ (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B)))
76anbi1i 434 . . . . 5 (((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) 𝐹:AB) ↔ ((Fun 𝐹 (𝐹 Fn A ran 𝐹 = B)) 𝐹:AB))
8 eqimss 2974 . . . . . . . 8 (ran 𝐹 = B → ran 𝐹B)
9 df-f 4833 . . . . . . . . 9 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
109biimpri 124 . . . . . . . 8 ((𝐹 Fn A ran 𝐹B) → 𝐹:AB)
118, 10sylan2 270 . . . . . . 7 ((𝐹 Fn A ran 𝐹 = B) → 𝐹:AB)
12113adant2 911 . . . . . 6 ((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) → 𝐹:AB)
1312pm4.71i 371 . . . . 5 ((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) ↔ ((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) 𝐹:AB))
14 ancom 253 . . . . 5 ((𝐹:AB (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B))) ↔ ((Fun 𝐹 (𝐹 Fn A ran 𝐹 = B)) 𝐹:AB))
157, 13, 143bitr4ri 202 . . . 4 ((𝐹:AB (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B))) ↔ (𝐹 Fn A Fun 𝐹 ran 𝐹 = B))
165, 15bitri 173 . . 3 (((𝐹:AB Fun 𝐹) (𝐹 Fn A ran 𝐹 = B)) ↔ (𝐹 Fn A Fun 𝐹 ran 𝐹 = B))
174, 16bitri 173 . 2 ((𝐹:A1-1B 𝐹:AontoB) ↔ (𝐹 Fn A Fun 𝐹 ran 𝐹 = B))
181, 17bitri 173 1 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A Fun 𝐹 ran 𝐹 = B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 873   = wceq 1228  wss 2894  ccnv 4271  ran crn 4273  Fun wfun 4823   Fn wfn 4824  wf 4825  1-1wf1 4826  ontowfo 4827  1-1-ontowf1o 4828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836
This theorem is referenced by:  dff1o3  5057  dff1o4  5059  f1orn  5061
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