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Mirrors > Home > MPE Home > Th. List > feqresmpt | Structured version Visualization version GIF version |
Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
feqmptd.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
feqresmpt.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
feqresmpt | ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feqmptd.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | feqresmpt.2 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
3 | 1, 2 | fssresd 5984 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
4 | 3 | feqmptd 6159 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
5 | fvres 6117 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
6 | 5 | mpteq2ia 4668 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
7 | 4, 6 | syl6eq 2660 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ⊆ wss 3540 ↦ cmpt 4643 ↾ cres 5040 ⟶wf 5800 ‘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-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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: pwfseqlem5 9364 swrd0val 13273 gsumpt 18184 dpjidcl 18280 regsumsupp 19787 tsmsxplem2 21767 dvmulbr 23508 dvlip 23560 lhop1lem 23580 loglesqrt 24299 jensenlem1 24513 jensen 24515 amgm 24517 gsumle 29110 coinflippv 29872 ftc1cnnclem 32653 dvasin 32666 dvacos 32667 dvreasin 32668 dvreacos 32669 areacirclem1 32670 itgperiod 38873 fourierdlem69 39068 fourierdlem73 39072 fourierdlem74 39073 fourierdlem75 39074 fourierdlem76 39075 fourierdlem81 39080 fourierdlem85 39084 fourierdlem88 39087 fourierdlem92 39091 fourierdlem97 39096 fourierdlem100 39099 fourierdlem101 39100 fourierdlem103 39102 fourierdlem104 39103 fourierdlem107 39106 fourierdlem111 39110 fourierdlem112 39111 fouriersw 39124 sge0tsms 39273 sge0resrnlem 39296 meadjiunlem 39358 omeunle 39406 isomenndlem 39420 pfxres 40251 ushgredgedga 40456 ushgredgedgaloop 40458 |
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