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| Mirrors > Home > MPE Home > Th. List > funforn | Structured version Visualization version 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 6596 | . 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | |
| 2 | dffn4 6826 | . 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 dom cdm 5685 ran crn 5686 Fun wfun 6555 Fn wfn 6556 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-fn 6564 df-fo 6567 |
| This theorem is referenced by: fimacnvinrn 7091 imacosupp 8234 ordtypelem8 9565 wdomima2g 9626 imadomg 10574 gruima 10842 oppglsm 19660 1stcrestlem 23460 dfac14 23626 qtoptop2 23707 fsupprnfi 32701 imadomfi 42003 rn1st 45280 |
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