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Theorem rlimres 15449
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴)

Proof of Theorem rlimres
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4192 . . . . . . . 8 (dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹
2 ssralv 4014 . . . . . . . 8 ((dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹 β†’ (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)))
31, 2ax-mp 5 . . . . . . 7 (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
43reximi 3084 . . . . . 6 (βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
54ralimi 3083 . . . . 5 (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
65anim2i 618 . . . 4 ((𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)) β†’ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)))
76a1i 11 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)) β†’ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
8 rlimf 15392 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ 𝐹:dom πΉβŸΆβ„‚)
9 rlimss 15393 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ dom 𝐹 βŠ† ℝ)
10 eqidd 2734 . . . 4 ((𝐹 β‡π‘Ÿ 𝐴 ∧ 𝑧 ∈ dom 𝐹) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘§))
118, 9, 10rlim 15386 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β‡π‘Ÿ 𝐴 ↔ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
12 fssres 6712 . . . . . 6 ((𝐹:dom πΉβŸΆβ„‚ ∧ (dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹) β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
138, 1, 12sylancl 587 . . . . 5 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
14 resres 5954 . . . . . . 7 ((𝐹 β†Ύ dom 𝐹) β†Ύ 𝐡) = (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡))
15 ffn 6672 . . . . . . . . 9 (𝐹:dom πΉβŸΆβ„‚ β†’ 𝐹 Fn dom 𝐹)
16 fnresdm 6624 . . . . . . . . 9 (𝐹 Fn dom 𝐹 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
1817reseq1d 5940 . . . . . . 7 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ dom 𝐹) β†Ύ 𝐡) = (𝐹 β†Ύ 𝐡))
1914, 18eqtr3id 2787 . . . . . 6 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)) = (𝐹 β†Ύ 𝐡))
2019feq1d 6657 . . . . 5 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚ ↔ (𝐹 β†Ύ 𝐡):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚))
2113, 20mpbid 231 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
221, 9sstrid 3959 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ (dom 𝐹 ∩ 𝐡) βŠ† ℝ)
23 elinel2 4160 . . . . . 6 (𝑧 ∈ (dom 𝐹 ∩ 𝐡) β†’ 𝑧 ∈ 𝐡)
2423fvresd 6866 . . . . 5 (𝑧 ∈ (dom 𝐹 ∩ 𝐡) β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘§) = (πΉβ€˜π‘§))
2524adantl 483 . . . 4 ((𝐹 β‡π‘Ÿ 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐡)) β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘§) = (πΉβ€˜π‘§))
2621, 22, 25rlim 15386 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴 ↔ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
277, 11, 263imtr4d 294 . 2 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴))
2827pm2.43i 52 1 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914   class class class wbr 5109  dom cdm 5637   β†Ύ cres 5639   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057  β„cr 11058   < clt 11197   ≀ cle 11198   βˆ’ cmin 11393  β„+crp 12923  abscabs 15128   β‡π‘Ÿ crli 15376
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-pm 8774  df-rlim 15380
This theorem is referenced by:  rlimres2  15452  pnt  26985
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