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| Mirrors > Home > MPE Home > Th. List > funfvima | Structured version Visualization version GIF version | ||
| Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
| Ref | Expression |
|---|---|
| funfvima | ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmres 6030 | . . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) | |
| 2 | 1 | elin2 4203 | . . . . . 6 ⊢ (𝐵 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹)) |
| 3 | funres 6608 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
| 4 | fvelrn 7096 | . . . . . . . . 9 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) | |
| 5 | 3, 4 | sylan 580 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) |
| 6 | df-ima 5698 | . . . . . . . . . 10 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
| 7 | 6 | eleq2i 2833 | . . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) |
| 8 | fvres 6925 | . . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐵) = (𝐹‘𝐵)) | |
| 9 | 8 | eleq1d 2826 | . . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴))) |
| 10 | 7, 9 | bitr4id 290 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴))) |
| 11 | 5, 10 | syl5ibrcom 247 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
| 12 | 11 | ex 412 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐵 ∈ dom (𝐹 ↾ 𝐴) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
| 13 | 2, 12 | biimtrrid 243 | . . . . 5 ⊢ (Fun 𝐹 → ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
| 14 | 13 | expd 415 | . . . 4 ⊢ (Fun 𝐹 → (𝐵 ∈ 𝐴 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))) |
| 15 | 14 | com12 32 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (Fun 𝐹 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))) |
| 16 | 15 | impd 410 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
| 17 | 16 | pm2.43b 55 | 1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 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-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: funfvima2 7251 elovimad 7481 tz7.48-2 8482 tz9.12lem3 9829 djuun 9966 swrdwrdsymb 14700 lindff1 21840 txcnp 23628 c1liplem1 26035 pthdivtx 29747 htthlem 30936 tpr2rico 33911 brsiga 34184 erdszelem8 35203 relowlpssretop 37365 limsuppnfdlem 45716 limsupresxr 45781 liminfresxr 45782 liminfvalxr 45798 |
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