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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 6544 | . . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
2 | 1 | anim1i 616 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹)) |
3 | df-fn 6500 | . . 3 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) | |
4 | df-ima 5647 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
5 | 4 | eqcomi 2746 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) |
6 | df-fo 6503 | . . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴))) | |
7 | 5, 6 | mpbiran2 709 | . . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴) Fn 𝐴) |
8 | ssdmres 5961 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐴) = 𝐴) | |
9 | 8 | anbi2i 624 | . . 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 233 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3911 dom cdm 5634 ran crn 5635 ↾ cres 5636 “ cima 5637 Fun wfun 6491 Fn wfn 6492 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-fo 6503 |
This theorem is referenced by: fimadmfoALT 6768 resdif 6806 f1oweALT 7906 imafiALT 9290 f1opwfi 9301 fodomfi2 9997 fin1a2lem7 10343 znnen 16095 connima 22779 1stcfb 22799 1stckgenlem 22907 qtoprest 23071 re2ndc 24167 uniiccdif 24945 opnmblALT 24970 mbfimaopnlem 25022 ffsrn 31649 cycpmconjvlem 31993 erdszelem2 33789 ivthALT 34810 poimirlem26 36107 poimirlem27 36108 lmhmfgima 41414 |
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