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Mirrors > Home > MPE Home > Th. List > fimadmfoALT | Structured version Visualization version GIF version |
Description: Alternate proof of fimadmfo 6814, based on fores 6815. A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fimadmfoALT | ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 6725 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | frel 6721 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
3 | resdm 6024 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
4 | 3 | eqcomd 2733 | . . . . 5 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
6 | reseq2 5974 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴)) | |
7 | 5, 6 | sylan9eq 2787 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹 ↾ 𝐴)) |
8 | 1, 7 | mpdan 686 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ 𝐴)) |
9 | ffun 6719 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
10 | eqimss2 4037 | . . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
12 | 9, 11 | jca 511 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
14 | fores 6815 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
16 | foeq1 6801 | . . . 4 ⊢ (𝐹 = (𝐹 ↾ 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) | |
17 | 16 | adantl 481 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) |
18 | 15, 17 | mpbird 257 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
19 | 8, 18 | 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 ⊆ wss 3944 dom cdm 5672 ↾ cres 5674 “ cima 5675 Rel wrel 5677 Fun wfun 6536 ⟶wf 6538 –onto→wfo 6540 |
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 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 |
This theorem is referenced by: (None) |
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