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| Mirrors > Home > MPE Home > Th. List > fvn0fvelrnOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of fvn0fvelrn 6937 as of 13-Jan-2025. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| fvn0fvelrnOLD | ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvfundmfvn0 6949 | . 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋}))) | |
| 2 | eldmressnsn 6042 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋})) | |
| 3 | fvelrn 7096 | . . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → ((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋})) | |
| 4 | pm3.2 469 | . . . . . . 7 ⊢ (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))) | |
| 5 | 3, 4 | syl 17 | . . . . . 6 ⊢ ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))) |
| 6 | 5 | ex 412 | . . . . 5 ⊢ (Fun (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))) |
| 7 | 6 | com13 88 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))) |
| 8 | 2, 7 | mpd 15 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))) |
| 9 | 8 | imp 406 | . 2 ⊢ ((𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)) |
| 10 | fvressn 7182 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋)) | |
| 11 | 10 | eleq1d 2826 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))) |
| 12 | fvrnressn 7181 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) | |
| 13 | 11, 12 | sylbid 240 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
| 14 | 13 | impcom 407 | . 2 ⊢ ((((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) ∈ ran 𝐹) |
| 15 | 1, 9, 14 | 3syl 18 | 1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 dom cdm 5685 ran crn 5686 ↾ cres 5687 Fun wfun 6555 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: (None) |
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