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Theorem cncfmptid 12496
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 3055 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
2 simpr 109 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
3 simpll 499 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑆𝑇)
4 simpr 109 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑆)
53, 4sseldd 3048 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑇)
65fmpttd 5507 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥):𝑆𝑇)
7 simpr 109 . . . 4 ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
87a1i 9 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+))
9 eqid 2100 . . . . . . . 8 (𝑥𝑆𝑥) = (𝑥𝑆𝑥)
10 id 19 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
11 simprll 507 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦𝑆)
129, 10, 11, 11fvmptd3 5446 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑦) = 𝑦)
13 id 19 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
14 simprlr 508 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑧𝑆)
159, 13, 14, 14fvmptd3 5446 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑧) = 𝑧)
1612, 15oveq12d 5724 . . . . . 6 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧)) = (𝑦𝑧))
1716fveq2d 5357 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) = (abs‘(𝑦𝑧)))
1817breq1d 3885 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤 ↔ (abs‘(𝑦𝑧)) < 𝑤))
1918exbiri 377 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+) → ((abs‘(𝑦𝑧)) < 𝑤 → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤)))
206, 8, 19elcncf1di 12479 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇)))
211, 2, 20mp2and 427 1 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1448  wss 3021   class class class wbr 3875  cmpt 3929  cfv 5059  (class class class)co 5706  cc 7498   < clt 7672  cmin 7804  +crp 9291  abscabs 10609  cnccncf 12470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474  df-cncf 12471
This theorem is referenced by:  expcncf  12504
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