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Theorem dffo2 5053
Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
dffo2 (𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))

Proof of Theorem dffo2
StepHypRef Expression
1 fof 5049 . . 3 (𝐹:AontoB𝐹:AB)
2 forn 5052 . . 3 (𝐹:AontoB → ran 𝐹 = B)
31, 2jca 290 . 2 (𝐹:AontoB → (𝐹:AB ran 𝐹 = B))
4 ffn 4989 . . 3 (𝐹:AB𝐹 Fn A)
5 df-fo 4851 . . . 4 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
65biimpri 124 . . 3 ((𝐹 Fn A ran 𝐹 = B) → 𝐹:AontoB)
74, 6sylan 267 . 2 ((𝐹:AB ran 𝐹 = B) → 𝐹:AontoB)
83, 7impbii 117 1 (𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  ran crn 4289   Fn wfn 4840  wf 4841  ontowfo 4843
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-fo 4851
This theorem is referenced by:  foco  5059  dff1o5  5078  dffo3  5257  dffo4  5258  fo1stresm  5730  fo2ndresm  5731  fo2ndf  5790  1fv  8766
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