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Mirrors > Home > MPE Home > Th. List > resfunexg | Structured version Visualization version GIF version |
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.) |
Ref | Expression |
---|---|
resfunexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 6596 | . . . . . . 7 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵)) | |
2 | 1 | adantr 479 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun (𝐴 ↾ 𝐵)) |
3 | 2 | funfnd 6585 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵)) |
4 | dffn5 6956 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥))) | |
5 | 3, 4 | sylib 217 | . . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥))) |
6 | fvex 6909 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵)‘𝑥) ∈ V | |
7 | 6 | fnasrn 7154 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) |
8 | 5, 7 | eqtrdi 2781 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉)) |
9 | opex 5466 | . . . . . 6 ⊢ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉 ∈ V | |
10 | eqid 2725 | . . . . . 6 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) | |
11 | 9, 10 | dmmpti 6700 | . . . . 5 ⊢ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) = dom (𝐴 ↾ 𝐵) |
12 | 11 | imaeq2i 6062 | . . . 4 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉)) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) |
13 | imadmrn 6074 | . . . 4 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) | |
14 | 12, 13 | eqtr3i 2755 | . . 3 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) |
15 | 8, 14 | eqtr4di 2783 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵))) |
16 | funmpt 6592 | . . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) | |
17 | dmresexg 6019 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V) | |
18 | 17 | adantl 480 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V) |
19 | funimaexg 6640 | . . 3 ⊢ ((Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) ∧ dom (𝐴 ↾ 𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) ∈ V) | |
20 | 16, 18, 19 | sylancr 585 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) ∈ V) |
21 | 15, 20 | eqeltrd 2825 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 〈cop 4636 ↦ cmpt 5232 dom cdm 5678 ran crn 5679 ↾ cres 5680 “ cima 5681 Fun wfun 6543 Fn wfn 6544 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 |
This theorem is referenced by: resiexd 7228 fnex 7229 ofexg 7690 cofunexg 7953 frrlem13 8304 naddcllem 8697 dfac8alem 10054 dfac12lem1 10168 cfsmolem 10295 alephsing 10301 itunifval 10441 zorn2lem1 10521 ttukeylem3 10536 imadomg 10559 wunex2 10763 inar1 10800 axdc4uzlem 13984 hashf1rn 14347 bpolylem 16028 1stf1 18186 1stf2 18187 2ndf1 18189 2ndf2 18190 1stfcl 18191 2ndfcl 18192 gsumzadd 19889 dfrngc2 20573 dfringc2 20602 rngcresringcat 20614 madeval 27825 addsval 27925 negsval 27984 mulsval 28059 satf 35091 tendo02 40387 dnnumch1 42607 aomclem6 42622 grimidvtxedg 47357 fdivval 47795 |
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