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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 6731 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | ffn 6722 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴) |
4 | dffn4 6817 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹) |
6 | imaeq2 6059 | . . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹)) | |
7 | imadmrn 6073 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
8 | 6, 7 | eqtrdi 2784 | . . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹) |
9 | 8 | eqcoms 2736 | . . . . 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 6809 | . . . 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 686 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 dom cdm 5678 ran crn 5679 “ cima 5681 Fn wfn 6543 ⟶wf 6544 –onto→wfo 6546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fn 6551 df-f 6552 df-fo 6554 |
This theorem is referenced by: wrdsymb 14525 imasmhm 33079 imasghm 33080 imasrhm 33081 imaslmhm 33082 r1pquslmic 33281 fundcmpsurinjimaid 46751 |
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