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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 5843 | . . . . . . 7 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵)) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun (𝐴 ↾ 𝐵)) |
3 | funfn 5833 | . . . . . 6 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵)) | |
4 | 2, 3 | sylib 207 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵)) |
5 | dffn5 6151 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥))) | |
6 | 4, 5 | sylib 207 | . . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥))) |
7 | fvex 6113 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵)‘𝑥) ∈ V | |
8 | 7 | fnasrn 6317 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) |
9 | 6, 8 | syl6eq 2660 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉)) |
10 | opex 4859 | . . . . . 6 ⊢ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉 ∈ V | |
11 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) | |
12 | 10, 11 | dmmpti 5936 | . . . . 5 ⊢ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) = dom (𝐴 ↾ 𝐵) |
13 | 12 | imaeq2i 5383 | . . . 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)) |
14 | imadmrn 5395 | . . . 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)〉) | |
15 | 13, 14 | eqtr3i 2634 | . . 3 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) |
16 | 9, 15 | syl6eqr 2662 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵))) |
17 | funmpt 5840 | . . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) | |
18 | dmresexg 5341 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V) | |
19 | 18 | adantl 481 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V) |
20 | funimaexg 5889 | . . 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) | |
21 | 17, 19, 20 | sylancr 694 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ 〈𝑥, ((𝐴 ↾ 𝐵)‘𝑥)〉) “ dom (𝐴 ↾ 𝐵)) ∈ V) |
22 | 16, 21 | eqeltrd 2688 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: resiexd 6385 fnex 6386 ofexg 6799 cofunexg 7023 dfac8alem 8735 dfac12lem1 8848 cfsmolem 8975 alephsing 8981 itunifval 9121 zorn2lem1 9201 ttukeylem3 9216 imadomg 9237 wunex2 9439 inar1 9476 axdc4uzlem 12644 hashf1rn 13004 hashf1rnOLD 13005 bpolylem 14618 1stf1 16655 1stf2 16656 2ndf1 16658 2ndf2 16659 1stfcl 16660 2ndfcl 16661 gsumzadd 18145 tendo02 35093 dnnumch1 36632 aomclem6 36647 1wlkreslem 40878 dfrngc2 41764 dfringc2 41810 rngcresringcat 41822 fdivval 42131 |
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