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Mirrors > Home > MPE Home > Th. List > o1res2 | Structured version Visualization version GIF version |
Description: The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.) |
Ref | Expression |
---|---|
rlimres2.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
o1res2.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝑂(1)) |
Ref | Expression |
---|---|
o1res2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimres2.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | resmptd 5908 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
3 | o1res2.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝑂(1)) | |
4 | o1res 14917 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝑂(1) → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ 𝑂(1)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ 𝑂(1)) |
6 | 2, 5 | eqeltrrd 2914 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 ↦ cmpt 5146 ↾ cres 5557 𝑂(1)co1 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-o1 14847 df-lo1 14848 |
This theorem is referenced by: logno1 25219 chebbnd2 26053 chpo1ub 26056 vmadivsumb 26059 vmalogdivsum2 26114 vmalogdivsum 26115 2vmadivsumlem 26116 selbergb 26125 selberg2lem 26126 selberg2b 26128 selberg3lem1 26133 selberg3lem2 26134 selberg3 26135 selberg4lem1 26136 selberg4 26137 pntrsumo1 26141 pntrlog2bndlem2 26154 pntrlog2bndlem4 26156 |
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