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Theorem cncnpi 22781
Description: A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnsscnp.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
cncnpi ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))

Proof of Theorem cncnpi
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnsscnp.1 . . . 4 𝑋 = βˆͺ 𝐽
2 eqid 2732 . . . 4 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnf 22749 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
43adantr 481 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
5 cnima 22768 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) β†’ (◑𝐹 β€œ 𝑦) ∈ 𝐽)
65ad2ant2r 745 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ (◑𝐹 β€œ 𝑦) ∈ 𝐽)
7 simpr 485 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
87adantr 481 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐴 ∈ 𝑋)
9 simprr 771 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ (πΉβ€˜π΄) ∈ 𝑦)
103ad2antrr 724 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
11 ffn 6717 . . . . . . 7 (𝐹:π‘‹βŸΆβˆͺ 𝐾 β†’ 𝐹 Fn 𝑋)
12 elpreima 7059 . . . . . . 7 (𝐹 Fn 𝑋 β†’ (𝐴 ∈ (◑𝐹 β€œ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (πΉβ€˜π΄) ∈ 𝑦)))
1310, 11, 123syl 18 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ (𝐴 ∈ (◑𝐹 β€œ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (πΉβ€˜π΄) ∈ 𝑦)))
148, 9, 13mpbir2and 711 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐴 ∈ (◑𝐹 β€œ 𝑦))
15 eqimss 4040 . . . . . . . 8 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))
1615biantrud 532 . . . . . . 7 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐴 ∈ π‘₯ ↔ (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))
17 eleq2 2822 . . . . . . 7 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ (◑𝐹 β€œ 𝑦)))
1816, 17bitr3d 280 . . . . . 6 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ ((𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦)) ↔ 𝐴 ∈ (◑𝐹 β€œ 𝑦)))
1918rspcev 3612 . . . . 5 (((◑𝐹 β€œ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◑𝐹 β€œ 𝑦)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦)))
206, 14, 19syl2anc 584 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦)))
2120expr 457 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π΄) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))
2221ralrimiva 3146 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))
23 cntop1 22743 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
2423adantr 481 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
251toptopon 22418 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
2624, 25sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
27 cntop2 22744 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
2827adantr 481 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ Top)
292toptopon 22418 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
3028, 29sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
31 iscnp3 22747 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆβˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))))
3226, 30, 7, 31syl3anc 1371 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆβˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))))
334, 22, 32mpbir2and 711 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  βˆͺ cuni 4908  β—‘ccnv 5675   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Topctop 22394  TopOnctopon 22411   Cn ccn 22727   CnP ccnp 22728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-top 22395  df-topon 22412  df-cn 22730  df-cnp 22731
This theorem is referenced by:  cnsscnp  22782  cncnp  22783  lmcn  22808  ptcn  23130  tmdcn2  23592  ghmcnp  23618  tsmsmhm  23649  tsmsadd  23650  dvcnp2  25436  dvaddbr  25454  dvmulbr  25455  dvcobr  25462  dvcjbr  25465  dvcnvlem  25492  lhop1lem  25529  dvcnvrelem2  25534  ftc1cn  25559  taylthlem2  25885  psercn  25937  abelth  25952  cxpcn3  26253  efrlim  26471  blocni  30053  cvmlift2lem11  34299  cvmlift2lem12  34300  cvmlift3lem7  34311  gg-dvcnp2  35169  gg-dvmulbr  35170  gg-dvcobr  35171  poimir  36516  ftc1cnnc  36555  cncfiooicclem1  44599  fouriercn  44938
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