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Mirrors > Home > ILE Home > Th. List > funforn | GIF version |
Description: A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
funforn | ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 5212 | . 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | |
2 | dffn4 5410 | . 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) | |
3 | 1, 2 | bitri 183 | 1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 dom cdm 4598 ran crn 4599 Fun wfun 5176 Fn wfn 5177 –onto→wfo 5180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1436 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-fn 5185 df-fo 5188 |
This theorem is referenced by: dvrecap 13224 |
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