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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 6758 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
4 | 3 | feqmptd 6960 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
5 | fvres 6910 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
6 | 5 | mpteq2ia 5251 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
7 | 4, 6 | eqtrdi 2788 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ⊆ wss 3948 ↦ cmpt 5231 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: pwfseqlem5 10657 pfxres 14628 gsumpt 19829 dpjidcl 19927 regsumsupp 21174 tsmsxplem2 23657 dvmulbr 25455 dvlip 25509 lhop1lem 25529 loglesqrt 26263 jensenlem1 26488 jensen 26490 amgm 26492 ushgredgedg 28483 ushgredgedgloop 28485 mgcf1o 32168 gsumzresunsn 32201 gsumpart 32202 gsumhashmul 32203 gsumle 32237 coinflippv 33477 fdvposlt 33606 fdvposle 33608 logdivsqrle 33657 gg-dvmulbr 35170 ftc1cnnclem 36554 dvasin 36567 dvacos 36568 dvreasin 36569 dvreacos 36570 areacirclem1 36571 dvrelog2 40924 dvrelog3 40925 cantnf2 42065 limsupvaluz2 44444 supcnvlimsup 44446 itgperiod 44687 fourierdlem69 44881 fourierdlem73 44885 fourierdlem74 44886 fourierdlem75 44887 fourierdlem76 44888 fourierdlem81 44893 fourierdlem85 44897 fourierdlem88 44900 fourierdlem92 44904 fourierdlem97 44909 fourierdlem100 44912 fourierdlem101 44913 fourierdlem103 44915 fourierdlem104 44916 fourierdlem107 44919 fourierdlem111 44923 fourierdlem112 44924 fouriersw 44937 sge0tsms 45086 sge0resrnlem 45109 meadjiunlem 45171 omeunle 45222 isomenndlem 45236 |
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