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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 4886 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 5340 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | fdm 5273 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
5 | 4 | imaeq2d 4876 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
6 | forn 5343 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
7 | 1, 5, 6 | 3eqtr3a 2194 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 dom cdm 4534 ran crn 4535 “ cima 4537 ⟶wf 5114 –onto→wfo 5116 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fn 5121 df-f 5122 df-fo 5124 |
This theorem is referenced by: foimacnv 5378 foima2 5646 fiintim 6810 fidcenumlemr 6836 |
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