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| Mirrors > Home > MPE Home > Th. List > fores | Structured version Visualization version GIF version | ||
| Description: Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
| Ref | Expression |
|---|---|
| fores | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funres 6608 | . . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
| 2 | 1 | anim1i 615 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹)) |
| 3 | df-fn 6564 | . . 3 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) | |
| 4 | df-ima 5698 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
| 5 | 4 | eqcomi 2746 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) |
| 6 | df-fo 6567 | . . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴))) | |
| 7 | 5, 6 | mpbiran2 710 | . . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴) Fn 𝐴) |
| 8 | ssdmres 6031 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐴) = 𝐴) | |
| 9 | 8 | anbi2i 623 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) |
| 10 | 3, 7, 9 | 3bitr4i 303 | . 2 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹)) |
| 11 | 2, 10 | sylibr 234 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊆ wss 3951 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Fun wfun 6555 Fn wfn 6556 –onto→wfo 6559 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-fo 6567 |
| This theorem is referenced by: fimadmfoALT 6831 resdif 6869 f1oweALT 7997 imafi 9353 f1opwfi 9396 fodomfi2 10100 fin1a2lem7 10446 znnen 16248 connima 23433 1stcfb 23453 1stckgenlem 23561 qtoprest 23725 re2ndc 24822 uniiccdif 25613 opnmblALT 25638 mbfimaopnlem 25690 ffsrn 32740 cycpmconjvlem 33161 erdszelem2 35197 ivthALT 36336 poimirlem26 37653 poimirlem27 37654 lmhmfgima 43096 |
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