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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 4784 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 5233 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | fdm 5166 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
5 | 4 | imaeq2d 4774 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
6 | forn 5236 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
7 | 1, 5, 6 | 3eqtr3a 2144 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 dom cdm 4438 ran crn 4439 “ cima 4441 ⟶wf 5011 –onto→wfo 5013 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-cnv 4446 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-fn 5018 df-f 5019 df-fo 5021 |
This theorem is referenced by: foimacnv 5271 foima2 5530 fiintim 6639 fidcenumlemr 6664 |
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