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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 6169 | . . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
2 | 1 | anim1i 608 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹)) |
3 | df-fn 6130 | . . 3 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) | |
4 | df-ima 5359 | . . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
5 | 4 | eqcomi 2834 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) |
6 | df-fo 6133 | . . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴))) | |
7 | 5, 6 | mpbiran2 701 | . . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴) Fn 𝐴) |
8 | ssdmres 5660 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐴) = 𝐴) | |
9 | 8 | anbi2i 616 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴)) |
10 | 3, 7, 9 | 3bitr4i 295 | . 2 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹)) |
11 | 2, 10 | sylibr 226 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ⊆ wss 3798 dom cdm 5346 ran crn 5347 ↾ cres 5348 “ cima 5349 Fun wfun 6121 Fn wfn 6122 –onto→wfo 6125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-res 5358 df-ima 5359 df-fun 6129 df-fn 6130 df-fo 6133 |
This theorem is referenced by: fimadmfoALT 6368 resdif 6402 f1oweALT 7417 imafi 8534 f1opwfi 8545 fodomfi2 9203 fin1a2lem7 9550 znnen 15322 connima 21606 1stcfb 21626 1stckgenlem 21734 qtoprest 21898 re2ndc 22981 uniiccdif 23751 opnmblALT 23776 mbfimaopnlem 23828 ffsrn 30048 erdszelem2 31716 ivthALT 32863 poimirlem26 33974 poimirlem27 33975 lmhmfgima 38492 |
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