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Mirrors > Home > ILE Home > Th. List > f1odm | GIF version |
Description: The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
f1odm | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofn 5415 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fndm 5269 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 dom cdm 4586 Fn wfn 5165 –1-1-onto→wf1o 5169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 |
This theorem depends on definitions: df-bi 116 df-fn 5173 df-f 5174 df-f1 5175 df-f1o 5177 |
This theorem is referenced by: f1imacnv 5431 f1opw2 6026 xpcomco 6771 mapen 6791 ssenen 6796 phplem4 6800 phplem4on 6812 dif1en 6824 fiintim 6873 caseinl 7035 caseinr 7036 ctssdccl 7055 fihasheqf1oi 10662 hashfacen 10707 fisumss 11289 |
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