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Mirrors > Home > MPE Home > Th. List > fimadmfoALT | Structured version Visualization version GIF version |
Description: Alternate proof of fimadmfo 6601, based on fores 6602. 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 6524 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | frel 6521 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
3 | resdm 5899 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
4 | 3 | eqcomd 2829 | . . . . 5 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
6 | reseq2 5850 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴)) | |
7 | 5, 6 | sylan9eq 2878 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹 ↾ 𝐴)) |
8 | 1, 7 | mpdan 685 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ 𝐴)) |
9 | ffun 6519 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
10 | eqimss2 4026 | . . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
12 | 9, 11 | jca 514 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) |
14 | fores 6602 | . . . 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 6588 | . . . 4 ⊢ (𝐹 = (𝐹 ↾ 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) | |
17 | 16 | adantl 484 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))) |
18 | 15, 17 | mpbird 259 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) |
19 | 8, 18 | 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 1537 ⊆ wss 3938 dom cdm 5557 ↾ cres 5559 “ cima 5560 Rel wrel 5562 Fun wfun 6351 ⟶wf 6353 –onto→wfo 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 |
This theorem is referenced by: (None) |
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