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Theorem cncfmptid 13377
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 3155 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
2 simpr 109 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
3 simpll 524 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑆𝑇)
4 simpr 109 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑆)
53, 4sseldd 3148 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑇)
65fmpttd 5651 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥):𝑆𝑇)
7 simpr 109 . . . 4 ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
87a1i 9 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+))
9 eqid 2170 . . . . . . . 8 (𝑥𝑆𝑥) = (𝑥𝑆𝑥)
10 id 19 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
11 simprll 532 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦𝑆)
129, 10, 11, 11fvmptd3 5589 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑦) = 𝑦)
13 id 19 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
14 simprlr 533 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑧𝑆)
159, 13, 14, 14fvmptd3 5589 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑧) = 𝑧)
1612, 15oveq12d 5871 . . . . . 6 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧)) = (𝑦𝑧))
1716fveq2d 5500 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) = (abs‘(𝑦𝑧)))
1817breq1d 3999 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤 ↔ (abs‘(𝑦𝑧)) < 𝑤))
1918exbiri 380 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+) → ((abs‘(𝑦𝑧)) < 𝑤 → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤)))
206, 8, 19elcncf1di 13360 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇)))
211, 2, 20mp2and 431 1 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2141  wss 3121   class class class wbr 3989  cmpt 4050  cfv 5198  (class class class)co 5853  cc 7772   < clt 7954  cmin 8090  +crp 9610  abscabs 10961  cnccncf 13351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628  df-cncf 13352
This theorem is referenced by:  expcncf  13386  dvcnp2cntop  13457
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