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Theorem dff1o5 5078
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o5 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B ran 𝐹 = B))

Proof of Theorem dff1o5
StepHypRef Expression
1 df-f1o 4852 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
2 f1f 5035 . . . . 5 (𝐹:A1-1B𝐹:AB)
32biantrurd 289 . . . 4 (𝐹:A1-1B → (ran 𝐹 = B ↔ (𝐹:AB ran 𝐹 = B)))
4 dffo2 5053 . . . 4 (𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))
53, 4syl6rbbr 188 . . 3 (𝐹:A1-1B → (𝐹:AontoB ↔ ran 𝐹 = B))
65pm5.32i 427 . 2 ((𝐹:A1-1B 𝐹:AontoB) ↔ (𝐹:A1-1B ran 𝐹 = B))
71, 6bitri 173 1 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B ran 𝐹 = B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  ran crn 4289  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-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:  f1orescnv  5085  frec2uzf1od  8873
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