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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 6714 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
4 | 3 | feqmptd 6915 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
5 | fvres 6866 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
6 | 5 | mpteq2ia 5213 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
7 | 4, 6 | eqtrdi 2793 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⊆ wss 3915 ↦ cmpt 5193 ↾ cres 5640 ⟶wf 6497 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 |
This theorem is referenced by: pwfseqlem5 10606 pfxres 14574 gsumpt 19746 dpjidcl 19844 regsumsupp 21042 tsmsxplem2 23521 dvmulbr 25319 dvlip 25373 lhop1lem 25393 loglesqrt 26127 jensenlem1 26352 jensen 26354 amgm 26356 ushgredgedg 28219 ushgredgedgloop 28221 mgcf1o 31905 gsumzresunsn 31938 gsumpart 31939 gsumhashmul 31940 gsumle 31974 coinflippv 33123 fdvposlt 33252 fdvposle 33254 logdivsqrle 33303 ftc1cnnclem 36178 dvasin 36191 dvacos 36192 dvreasin 36193 dvreacos 36194 areacirclem1 36195 dvrelog2 40550 dvrelog3 40551 cantnf2 41689 limsupvaluz2 44053 supcnvlimsup 44055 itgperiod 44296 fourierdlem69 44490 fourierdlem73 44494 fourierdlem74 44495 fourierdlem75 44496 fourierdlem76 44497 fourierdlem81 44502 fourierdlem85 44506 fourierdlem88 44509 fourierdlem92 44513 fourierdlem97 44518 fourierdlem100 44521 fourierdlem101 44522 fourierdlem103 44524 fourierdlem104 44525 fourierdlem107 44528 fourierdlem111 44532 fourierdlem112 44533 fouriersw 44546 sge0tsms 44695 sge0resrnlem 44718 meadjiunlem 44780 omeunle 44831 isomenndlem 44845 |
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