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| Mirrors > Home > MPE Home > Th. List > fimadmfo | Structured version Visualization version 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 6745 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
| 2 | ffn 6736 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴) |
| 4 | dffn4 6826 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
| 5 | 3, 4 | sylib 218 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹) |
| 6 | imaeq2 6074 | . . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹)) | |
| 7 | imadmrn 6088 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
| 8 | 6, 7 | eqtrdi 2793 | . . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹) |
| 9 | 8 | eqcoms 2745 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
| 10 | 9 | adantl 481 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹 “ 𝐴) = ran 𝐹) |
| 11 | foeq3 6818 | . . . 4 ⊢ ((𝐹 “ 𝐴) = ran 𝐹 → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹)) | |
| 12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹)) |
| 13 | 5, 12 | mpbird 257 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
| 14 | 1, 13 | mpdan 687 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 dom cdm 5685 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fn 6564 df-f 6565 df-fo 6567 |
| This theorem is referenced by: wrdsymb 14580 imasmhm 33382 imasghm 33383 imasrhm 33384 imaslmhm 33385 r1pquslmic 33631 fundcmpsurinjimaid 47398 |
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