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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 6610 | . . 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 6566 | . . 3 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) | |
4 | df-ima 5702 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
5 | 4 | eqcomi 2744 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) |
6 | df-fo 6569 | . . . 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 6033 | . . . 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 1537 ⊆ wss 3963 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Fun wfun 6557 Fn wfn 6558 –onto→wfo 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-fo 6569 |
This theorem is referenced by: fimadmfoALT 6832 resdif 6870 f1oweALT 7996 imafi 9351 f1opwfi 9394 fodomfi2 10098 fin1a2lem7 10444 znnen 16245 connima 23449 1stcfb 23469 1stckgenlem 23577 qtoprest 23741 re2ndc 24837 uniiccdif 25627 opnmblALT 25652 mbfimaopnlem 25704 ffsrn 32747 cycpmconjvlem 33144 erdszelem2 35177 ivthALT 36318 poimirlem26 37633 poimirlem27 37634 lmhmfgima 43073 |
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