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Mirrors > Home > ILE Home > Th. List > fimadmfo | GIF version |
Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) |
Ref | Expression |
---|---|
fimadmfo | ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5401 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | ffn 5395 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴) |
4 | dffn4 5474 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
5 | 3, 4 | sylib 122 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹) |
6 | imaeq2 4995 | . . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹)) | |
7 | imadmrn 5009 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
8 | 6, 7 | eqtrdi 2242 | . . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹) |
9 | 8 | eqcoms 2196 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
10 | 9 | adantl 277 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹 “ 𝐴) = ran 𝐹) |
11 | foeq3 5466 | . . . 4 ⊢ ((𝐹 “ 𝐴) = ran 𝐹 → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹)) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹)) |
13 | 5, 12 | mpbird 167 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
14 | 1, 13 | mpdan 421 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 dom cdm 4655 ran crn 4656 “ cima 4658 Fn wfn 5241 ⟶wf 5242 –onto→wfo 5244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-xp 4661 df-cnv 4663 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-fn 5249 df-f 5250 df-fo 5252 |
This theorem is referenced by: wrdsymb 10931 |
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