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Theorem cncfmptid 14486
Description: The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
Assertion
Ref Expression
cncfmptid ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem cncfmptid
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr 3178 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
2 simpr 110 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
3 simpll 527 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑆𝑇)
4 simpr 110 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑆)
53, 4sseldd 3171 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑇)
65fmpttd 5687 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥):𝑆𝑇)
7 simpr 110 . . . 4 ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
87a1i 9 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+))
9 eqid 2189 . . . . . . . 8 (𝑥𝑆𝑥) = (𝑥𝑆𝑥)
10 id 19 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
11 simprll 537 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦𝑆)
129, 10, 11, 11fvmptd3 5625 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑦) = 𝑦)
13 id 19 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
14 simprlr 538 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑧𝑆)
159, 13, 14, 14fvmptd3 5625 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑧) = 𝑧)
1612, 15oveq12d 5909 . . . . . 6 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧)) = (𝑦𝑧))
1716fveq2d 5534 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) = (abs‘(𝑦𝑧)))
1817breq1d 4028 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤 ↔ (abs‘(𝑦𝑧)) < 𝑤))
1918exbiri 382 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+) → ((abs‘(𝑦𝑧)) < 𝑤 → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤)))
206, 8, 19elcncf1di 14469 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇)))
211, 2, 20mp2and 433 1 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  wss 3144   class class class wbr 4018  cmpt 4079  cfv 5231  (class class class)co 5891  cc 7828   < clt 8011  cmin 8147  +crp 9672  abscabs 11025  cnccncf 14460
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-map 6668  df-cncf 14461
This theorem is referenced by:  expcncf  14495  dvcnp2cntop  14566
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