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Theorem dff1o2 5074
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 4852 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
2 df-f1 4850 . . . 4 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
3 df-fo 4851 . . . 4 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
42, 3anbi12i 433 . . 3 ((𝐹:A1-1B 𝐹:AontoB) ↔ ((𝐹:AB Fun 𝐹) (𝐹 Fn A ran 𝐹 = B)))
5 anass 381 . . . 4 (((𝐹:AB Fun 𝐹) (𝐹 Fn A ran 𝐹 = B)) ↔ (𝐹:AB (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B))))
6 3anan12 896 . . . . . 6 ((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) ↔ (Fun 𝐹 (𝐹 Fn A ran 𝐹 = B)))
76anbi1i 431 . . . . 5 (((𝐹 Fn A Fun 𝐹 ran 𝐹 = B) 𝐹:AB) ↔ ((Fun 𝐹 (𝐹 Fn A ran 𝐹 = B)) 𝐹:AB))
8 eqimss 2991 . . . . . . . 8 (ran 𝐹 = B → ran 𝐹B)
9 df-f 4849 . . . . . . . . 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 922 . . . . . 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 884   = wceq 1242  wss 2911  ccnv 4287  ran crn 4289  Fun wfun 4839   Fn wfn 4840  wf 4841  1-1wf1 4842  ontowfo 4843  1-1-ontowf1o 4844
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  dff1o3  5075  dff1o4  5077  f1orn  5079
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