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Theorem fof 5049
 Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fof (𝐹:AontoB𝐹:AB)

Proof of Theorem fof
StepHypRef Expression
1 eqimss 2991 . . 3 (ran 𝐹 = B → ran 𝐹B)
21anim2i 324 . 2 ((𝐹 Fn A ran 𝐹 = B) → (𝐹 Fn A ran 𝐹B))
3 df-fo 4851 . 2 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
4 df-f 4849 . 2 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
52, 3, 43imtr4i 190 1 (𝐹:AontoB𝐹:AB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841  –onto→wfo 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:  fofun  5050  fofn  5051  dffo2  5053  foima  5054  resdif  5091  ffoss  5101  fconstfvm  5322  cocan2  5371  foeqcnvco  5373  fornex  5684  algrflem  5792  tposf2  5824  ssdomg  6194  fopwdom  6246
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