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Theorem limcresi 14174
Description: Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)
Assertion
Ref Expression
limcresi (𝐹 limβ„‚ 𝐡) βŠ† ((𝐹 β†Ύ 𝐢) limβ„‚ 𝐡)

Proof of Theorem limcresi
Dummy variables 𝑑 𝑒 𝑒 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limcrcl 14166 . . . . . . 7 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (𝐹:dom πΉβŸΆβ„‚ ∧ dom 𝐹 βŠ† β„‚ ∧ 𝐡 ∈ β„‚))
21simp1d 1009 . . . . . 6 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ 𝐹:dom πΉβŸΆβ„‚)
31simp2d 1010 . . . . . 6 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ dom 𝐹 βŠ† β„‚)
41simp3d 1011 . . . . . 6 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ 𝐡 ∈ β„‚)
52, 3, 4ellimc3ap 14169 . . . . 5 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ↔ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒))))
65ibi 176 . . . 4 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
7 inss1 3357 . . . . . . . . 9 (dom 𝐹 ∩ 𝐢) βŠ† dom 𝐹
8 ssralv 3221 . . . . . . . . 9 ((dom 𝐹 ∩ 𝐢) βŠ† dom 𝐹 β†’ (βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
97, 8ax-mp 5 . . . . . . . 8 (βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒))
10 elinel2 3324 . . . . . . . . . . . . . . 15 (𝑒 ∈ (dom 𝐹 ∩ 𝐢) β†’ 𝑒 ∈ 𝐢)
11 fvres 5541 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝐢 β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘’) = (πΉβ€˜π‘’))
1210, 11syl 14 . . . . . . . . . . . . . 14 (𝑒 ∈ (dom 𝐹 ∩ 𝐢) β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘’) = (πΉβ€˜π‘’))
1312adantl 277 . . . . . . . . . . . . 13 ((π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ∧ 𝑒 ∈ (dom 𝐹 ∩ 𝐢)) β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘’) = (πΉβ€˜π‘’))
1413fvoveq1d 5899 . . . . . . . . . . . 12 ((π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ∧ 𝑒 ∈ (dom 𝐹 ∩ 𝐢)) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) = (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)))
1514breq1d 4015 . . . . . . . . . . 11 ((π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ∧ 𝑒 ∈ (dom 𝐹 ∩ 𝐢)) β†’ ((absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒 ↔ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒))
1615imbi2d 230 . . . . . . . . . 10 ((π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ∧ 𝑒 ∈ (dom 𝐹 ∩ 𝐢)) β†’ (((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒) ↔ ((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
1716biimprd 158 . . . . . . . . 9 ((π‘₯ ∈ (𝐹 limβ„‚ 𝐡) ∧ 𝑒 ∈ (dom 𝐹 ∩ 𝐢)) β†’ (((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ ((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
1817ralimdva 2544 . . . . . . . 8 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
199, 18syl5 32 . . . . . . 7 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
2019reximdv 2578 . . . . . 6 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
2120ralimdv 2545 . . . . 5 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒) β†’ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
2221anim2d 337 . . . 4 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ ((π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom 𝐹((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((πΉβ€˜π‘’) βˆ’ π‘₯)) < 𝑒)) β†’ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒))))
236, 22mpd 13 . . 3 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒)))
24 fresin 5396 . . . . 5 (𝐹:dom πΉβŸΆβ„‚ β†’ (𝐹 β†Ύ 𝐢):(dom 𝐹 ∩ 𝐢)βŸΆβ„‚)
252, 24syl 14 . . . 4 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (𝐹 β†Ύ 𝐢):(dom 𝐹 ∩ 𝐢)βŸΆβ„‚)
267, 3sstrid 3168 . . . 4 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (dom 𝐹 ∩ 𝐢) βŠ† β„‚)
2725, 26, 4ellimc3ap 14169 . . 3 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ (π‘₯ ∈ ((𝐹 β†Ύ 𝐢) limβ„‚ 𝐡) ↔ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ (dom 𝐹 ∩ 𝐢)((𝑒 # 𝐡 ∧ (absβ€˜(𝑒 βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜(((𝐹 β†Ύ 𝐢)β€˜π‘’) βˆ’ π‘₯)) < 𝑒))))
2823, 27mpbird 167 . 2 (π‘₯ ∈ (𝐹 limβ„‚ 𝐡) β†’ π‘₯ ∈ ((𝐹 β†Ύ 𝐢) limβ„‚ 𝐡))
2928ssriv 3161 1 (𝐹 limβ„‚ 𝐡) βŠ† ((𝐹 β†Ύ 𝐢) limβ„‚ 𝐡)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456   ∩ cin 3130   βŠ† wss 3131   class class class wbr 4005  dom cdm 4628   β†Ύ cres 4630  βŸΆwf 5214  β€˜cfv 5218  (class class class)co 5877  β„‚cc 7811   < clt 7994   βˆ’ cmin 8130   # cap 8540  β„+crp 9655  abscabs 11008   limβ„‚ climc 14162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pm 6653  df-limced 14164
This theorem is referenced by:  dvidlemap  14199  dvcnp2cntop  14202  dvcoapbr  14210
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