MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncnpi Structured version   Visualization version   GIF version

Theorem cncnpi 23002
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 22970 . . 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 22989 . . . . . 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 22964 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
2423adantr 481 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
251toptopon 22639 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
2624, 25sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
27 cntop2 22965 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
2827adantr 481 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ Top)
292toptopon 22639 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
3028, 29sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
31 iscnp3 22968 . . 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 7411  Topctop 22615  TopOnctopon 22632   Cn ccn 22948   CnP ccnp 22949
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 7727
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 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-top 22616  df-topon 22633  df-cn 22951  df-cnp 22952
This theorem is referenced by:  cnsscnp  23003  cncnp  23004  lmcn  23029  ptcn  23351  tmdcn2  23813  ghmcnp  23839  tsmsmhm  23870  tsmsadd  23871  dvcnp2  25661  dvaddbr  25679  dvmulbr  25680  dvcobr  25687  dvcjbr  25690  dvcnvlem  25717  lhop1lem  25754  dvcnvrelem2  25759  ftc1cn  25784  taylthlem2  26110  psercn  26162  abelth  26177  cxpcn3  26480  efrlim  26698  blocni  30313  cvmlift2lem11  34590  cvmlift2lem12  34591  cvmlift3lem7  34602  gg-dvcnp2  35460  gg-dvmulbr  35461  gg-dvcobr  35462  poimir  36824  ftc1cnnc  36863  cncfiooicclem1  44908  fouriercn  45247
  Copyright terms: Public domain W3C validator