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Theorem iscnp 22963
Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscnp ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝐾,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝐹,𝑦   π‘₯,𝑃,𝑦   π‘₯,π‘Œ,𝑦
Proof of Theorem iscnp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnpval 22962 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐽 CnP 𝐾)β€˜π‘ƒ) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
21eleq2d 2817 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ 𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))}))
3 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘ƒ) = (πΉβ€˜π‘ƒ))
43eleq1d 2816 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘ƒ) ∈ 𝑦 ↔ (πΉβ€˜π‘ƒ) ∈ 𝑦))
5 imaeq1 6055 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (𝑓 β€œ π‘₯) = (𝐹 β€œ π‘₯))
65sseq1d 4014 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((𝑓 β€œ π‘₯) βŠ† 𝑦 ↔ (𝐹 β€œ π‘₯) βŠ† 𝑦))
76anbi2d 627 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦) ↔ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
87rexbidv 3176 . . . . . . 7 (𝑓 = 𝐹 β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
94, 8imbi12d 343 . . . . . 6 (𝑓 = 𝐹 β†’ (((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)) ↔ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))))
109ralbidv 3175 . . . . 5 (𝑓 = 𝐹 β†’ (βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)) ↔ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))))
1110elrab 3684 . . . 4 (𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))} ↔ (𝐹 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))))
12 toponmax 22650 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
13 toponmax 22650 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
14 elmapg 8837 . . . . . 6 ((π‘Œ ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1512, 13, 14syl2anr 595 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1615anbi1d 628 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
1711, 16bitrid 282 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))} ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
18173adant3 1130 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))} ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
192, 18bitrd 278 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   βŠ† wss 3949   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413   ↑m cmap 8824  TopOnctopon 22634   CnP ccnp 22951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-top 22618  df-topon 22635  df-cnp 22954
This theorem is referenced by:  iscnp2  22965  iscnp3  22970  tgcnp  22979  iscnp4  22989  cnconst2  23009  cnpresti  23014  cnprest  23015  cnprest2  23016  1stccnp  23188  cnpflf2  23726  symgtgp  23832  ghmcnp  23841  ellimc2  25628  xrlimcnp  26707  poimir  36826  icccncfext  44903
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