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Mirrors > Home > NFE Home > Th. List > f1odm | GIF version |
Description: The domain of a one-to-one onto mapping. (Contributed by set.mm contributors, 8-Mar-2014.) |
Ref | Expression |
---|---|
f1odm | ⊢ (F:A–1-1-onto→B → dom F = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofn 5288 | . 2 ⊢ (F:A–1-1-onto→B → F Fn A) | |
2 | fndm 5182 | . 2 ⊢ (F Fn A → dom F = A) | |
3 | 1, 2 | syl 15 | 1 ⊢ (F:A–1-1-onto→B → dom F = A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 dom cdm 4772 Fn wfn 4776 –1-1-onto→wf1o 4780 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 |
This theorem depends on definitions: df-bi 177 df-an 360 df-fn 4790 df-f 4791 df-f1 4792 df-f1o 4794 |
This theorem is referenced by: bren 6030 enpw1 6062 enmap1lem5 6073 nenpw1pwlem2 6085 ncdisjun 6136 sbthlem3 6205 |
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