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Mirrors > Home > MPE Home > Th. List > fimadmfoALT | Structured version Visualization version GIF version |
Description: Alternate proof of fimadmfo 6574, based on fores 6575. 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 6495 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | frel 6492 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
3 | resdm 5863 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
4 | 3 | eqcomd 2804 | . . . . 5 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
6 | reseq2 5813 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴)) | |
7 | 5, 6 | sylan9eq 2853 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹 ↾ 𝐴)) |
8 | 1, 7 | mpdan 686 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ 𝐴)) |
9 | ffun 6490 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
10 | eqimss2 3972 | . . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
12 | 9, 11 | jca 515 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
14 | fores 6575 | . . . 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 6561 | . . . 4 ⊢ (𝐹 = (𝐹 ↾ 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) | |
17 | 16 | adantl 485 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) |
18 | 15, 17 | mpbird 260 | . 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 209 ∧ wa 399 = wceq 1538 ⊆ wss 3881 dom cdm 5519 ↾ cres 5521 “ cima 5522 Rel wrel 5524 Fun wfun 6318 ⟶wf 6320 –onto→wfo 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 |
This theorem is referenced by: (None) |
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