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Mirrors > Home > ILE Home > Th. List > foima | GIF version |
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
Ref | Expression |
---|---|
foima | ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 4899 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 5353 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | fdm 5286 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
5 | 4 | imaeq2d 4889 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
6 | forn 5356 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
7 | 1, 5, 6 | 3eqtr3a 2197 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 dom cdm 4547 ran crn 4548 “ cima 4550 ⟶wf 5127 –onto→wfo 5129 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fn 5134 df-f 5135 df-fo 5137 |
This theorem is referenced by: foimacnv 5393 foima2 5661 fiintim 6825 fidcenumlemr 6851 |
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