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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 6515 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | ffn 6507 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴) |
4 | dffn4 6589 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
5 | 3, 4 | sylib 220 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹) |
6 | imaeq2 5918 | . . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹)) | |
7 | imadmrn 5932 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
8 | 6, 7 | syl6eq 2871 | . . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹) |
9 | 8 | eqcoms 2828 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
10 | 9 | adantl 484 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹 “ 𝐴) = ran 𝐹) |
11 | foeq3 6581 | . . . 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 259 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
14 | 1, 13 | mpdan 685 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 dom cdm 5548 ran crn 5549 “ cima 5551 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fn 6351 df-f 6352 df-fo 6354 |
This theorem is referenced by: wrdsymb 13886 fundcmpsurinjimaid 43646 |
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