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Theorem cncfmptid 14323
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 3175 . 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 3168 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑇)
65fmpttd 5684 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥):𝑆𝑇)
7 simpr 110 . . . 4 ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
87a1i 9 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+))
9 eqid 2187 . . . . . . . 8 (𝑥𝑆𝑥) = (𝑥𝑆𝑥)
10 id 19 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
11 simprll 537 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦𝑆)
129, 10, 11, 11fvmptd3 5622 . . . . . . 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 5622 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑧) = 𝑧)
1612, 15oveq12d 5906 . . . . . 6 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧)) = (𝑦𝑧))
1716fveq2d 5531 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) = (abs‘(𝑦𝑧)))
1817breq1d 4025 . . . 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 14306 . 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 2158  wss 3141   class class class wbr 4015  cmpt 4076  cfv 5228  (class class class)co 5888  cc 7822   < clt 8005  cmin 8141  +crp 9666  abscabs 11019  cnccncf 14297
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-map 6663  df-cncf 14298
This theorem is referenced by:  expcncf  14332  dvcnp2cntop  14403
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