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Theorem cnpfval 22958
Description: The function mapping the points in a topology 𝐽 to the set of all functions from 𝐽 to topology 𝐾 continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnpfval ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
Distinct variable groups:   𝑀,𝑓,π‘₯,𝐾   𝑓,𝑋,𝑀,π‘₯   𝑓,π‘Œ,𝑀,π‘₯   𝑣,𝑓,𝐽,𝑀,π‘₯
Allowed substitution hints:   𝐾(𝑣)   𝑋(𝑣)   π‘Œ(𝑣)

Proof of Theorem cnpfval
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnp 22952 . . 3 CnP = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ ↑m βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
21a1i 11 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ CnP = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ ↑m βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))})))
3 simprl 767 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑗 = 𝐽)
43unieqd 4921 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
5 toponuni 22636 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65ad2antrr 722 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑋 = βˆͺ 𝐽)
74, 6eqtr4d 2773 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = 𝑋)
8 simprr 769 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘˜ = 𝐾)
98unieqd 4921 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
10 toponuni 22636 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
1110ad2antlr 723 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘Œ = βˆͺ 𝐾)
129, 11eqtr4d 2773 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = π‘Œ)
1312, 7oveq12d 7429 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆͺ π‘˜ ↑m βˆͺ 𝑗) = (π‘Œ ↑m 𝑋))
143rexeqdv 3324 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀) ↔ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)))
1514imbi2d 339 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)) ↔ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))))
168, 15raleqbidv 3340 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)) ↔ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))))
1713, 16rabeqbidv 3447 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ {𝑓 ∈ (βˆͺ π‘˜ ↑m βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))})
187, 17mpteq12dv 5238 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ ↑m βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
19 topontop 22635 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2019adantr 479 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ Top)
21 topontop 22635 . . 3 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2221adantl 480 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ Top)
23 ovex 7444 . . . . . 6 (π‘Œ ↑m 𝑋) ∈ V
24 ssrab2 4076 . . . . . 6 {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} βŠ† (π‘Œ ↑m 𝑋)
2523, 24elpwi2 5345 . . . . 5 {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} ∈ 𝒫 (π‘Œ ↑m 𝑋)
2625a1i 11 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} ∈ 𝒫 (π‘Œ ↑m 𝑋))
2726fmpttd 7115 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}):π‘‹βŸΆπ’« (π‘Œ ↑m 𝑋))
28 toponmax 22648 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2928adantr 479 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑋 ∈ 𝐽)
3023pwex 5377 . . . 4 𝒫 (π‘Œ ↑m 𝑋) ∈ V
3130a1i 11 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝒫 (π‘Œ ↑m 𝑋) ∈ V)
32 fex2 7926 . . 3 (((π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}):π‘‹βŸΆπ’« (π‘Œ ↑m 𝑋) ∧ 𝑋 ∈ 𝐽 ∧ 𝒫 (π‘Œ ↑m 𝑋) ∈ V) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) ∈ V)
3327, 29, 31, 32syl3anc 1369 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) ∈ V)
342, 18, 20, 22, 33ovmpod 7562 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   ↦ cmpt 5230   β€œ cima 5678  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413   ↑m cmap 8822  Topctop 22615  TopOnctopon 22632   CnP ccnp 22949
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
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 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-top 22616  df-topon 22633  df-cnp 22952
This theorem is referenced by:  cnpval  22960  iscnp2  22963  cnambfre  36839
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