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Mirrors > Home > MPE Home > Th. List > lo1res2 | Structured version Visualization version GIF version |
Description: The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
rlimres2.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
lo1res2.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ ≤𝑂(1)) |
Ref | Expression |
---|---|
lo1res2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimres2.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | resmptd 5910 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
3 | lo1res2.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ ≤𝑂(1)) | |
4 | lo1res 14918 | . . 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 2916 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3938 ↦ cmpt 5148 ↾ cres 5559 ≤𝑂(1)clo1 14846 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ico 12747 df-lo1 14850 |
This theorem is referenced by: (None) |
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