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Theorem cncnpi 22782
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 2733 . . . 4 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnf 22750 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
43adantr 482 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
5 cnima 22769 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) β†’ (◑𝐹 β€œ 𝑦) ∈ 𝐽)
65ad2ant2r 746 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ (◑𝐹 β€œ 𝑦) ∈ 𝐽)
7 simpr 486 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
87adantr 482 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐴 ∈ 𝑋)
9 simprr 772 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ (πΉβ€˜π΄) ∈ 𝑦)
103ad2antrr 725 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
11 ffn 6718 . . . . . . 7 (𝐹:π‘‹βŸΆβˆͺ 𝐾 β†’ 𝐹 Fn 𝑋)
12 elpreima 7060 . . . . . . 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 712 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ 𝑦)) β†’ 𝐴 ∈ (◑𝐹 β€œ 𝑦))
15 eqimss 4041 . . . . . . . 8 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))
1615biantrud 533 . . . . . . 7 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐴 ∈ π‘₯ ↔ (𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦))))
17 eleq2 2823 . . . . . . 7 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ (◑𝐹 β€œ 𝑦)))
1816, 17bitr3d 281 . . . . . 6 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ ((𝐴 ∈ π‘₯ ∧ π‘₯ βŠ† (◑𝐹 β€œ 𝑦)) ↔ 𝐴 ∈ (◑𝐹 β€œ 𝑦)))
1918rspcev 3613 . . . . 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 585 . . . 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 458 . . 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 3147 . 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 22744 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
2423adantr 482 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
251toptopon 22419 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
2624, 25sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
27 cntop2 22745 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
2827adantr 482 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ Top)
292toptopon 22419 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
3028, 29sylib 217 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
31 iscnp3 22748 . . 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 1372 . 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 712 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  βˆͺ cuni 4909  β—‘ccnv 5676   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412   Cn ccn 22728   CnP ccnp 22729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-cnp 22732
This theorem is referenced by:  cnsscnp  22783  cncnp  22784  lmcn  22809  ptcn  23131  tmdcn2  23593  ghmcnp  23619  tsmsmhm  23650  tsmsadd  23651  dvcnp2  25437  dvaddbr  25455  dvmulbr  25456  dvcobr  25463  dvcjbr  25466  dvcnvlem  25493  lhop1lem  25530  dvcnvrelem2  25535  ftc1cn  25560  taylthlem2  25886  psercn  25938  abelth  25953  cxpcn3  26256  efrlim  26474  blocni  30058  cvmlift2lem11  34304  cvmlift2lem12  34305  cvmlift3lem7  34316  gg-dvcnp2  35174  gg-dvmulbr  35175  gg-dvcobr  35176  poimir  36521  ftc1cnnc  36560  cncfiooicclem1  44609  fouriercn  44948
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