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Theorem iscnp4 23187
Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃 " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
iscnp4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦)))
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝐽,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝑃,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦

Proof of Theorem iscnp4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cnpf2 23174 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1115 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1165 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 simplr 767 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))
5 simpll2 1210 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
6 topontop 22835 . . . . . . . . 9 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
75, 6syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝐾 ∈ Top)
8 eqid 2728 . . . . . . . . . 10 βˆͺ 𝐾 = βˆͺ 𝐾
98neii1 23030 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝑦 βŠ† βˆͺ 𝐾)
107, 9sylancom 586 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝑦 βŠ† βˆͺ 𝐾)
118ntropn 22973 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑦 βŠ† βˆͺ 𝐾) β†’ ((intβ€˜πΎ)β€˜π‘¦) ∈ 𝐾)
127, 10, 11syl2anc 582 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ ((intβ€˜πΎ)β€˜π‘¦) ∈ 𝐾)
13 simpr 483 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}))
143adantr 479 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
15 simpll3 1211 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ 𝑃 ∈ 𝑋)
1614, 15ffvelcdmd 7100 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ (πΉβ€˜π‘ƒ) ∈ π‘Œ)
17 toponuni 22836 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
185, 17syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ π‘Œ = βˆͺ 𝐾)
1916, 18eleqtrd 2831 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ (πΉβ€˜π‘ƒ) ∈ βˆͺ 𝐾)
2019snssd 4817 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ {(πΉβ€˜π‘ƒ)} βŠ† βˆͺ 𝐾)
218neiint 23028 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ {(πΉβ€˜π‘ƒ)} βŠ† βˆͺ 𝐾 ∧ 𝑦 βŠ† βˆͺ 𝐾) β†’ (𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) ↔ {(πΉβ€˜π‘ƒ)} βŠ† ((intβ€˜πΎ)β€˜π‘¦)))
227, 20, 10, 21syl3anc 1368 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ (𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) ↔ {(πΉβ€˜π‘ƒ)} βŠ† ((intβ€˜πΎ)β€˜π‘¦)))
2313, 22mpbid 231 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ {(πΉβ€˜π‘ƒ)} βŠ† ((intβ€˜πΎ)β€˜π‘¦))
24 fvex 6915 . . . . . . . . 9 (πΉβ€˜π‘ƒ) ∈ V
2524snss 4794 . . . . . . . 8 ((πΉβ€˜π‘ƒ) ∈ ((intβ€˜πΎ)β€˜π‘¦) ↔ {(πΉβ€˜π‘ƒ)} βŠ† ((intβ€˜πΎ)β€˜π‘¦))
2623, 25sylibr 233 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ (πΉβ€˜π‘ƒ) ∈ ((intβ€˜πΎ)β€˜π‘¦))
27 cnpimaex 23180 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ ((intβ€˜πΎ)β€˜π‘¦) ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ ((intβ€˜πΎ)β€˜π‘¦)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))
284, 12, 26, 27syl3anc 1368 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))
29 simpl1 1188 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3029ad2antrr 724 . . . . . . . 8 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
31 topontop 22835 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3230, 31syl 17 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ 𝐽 ∈ Top)
33 simprl 769 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ π‘₯ ∈ 𝐽)
34 simprrl 779 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ 𝑃 ∈ π‘₯)
35 opnneip 23043 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}))
3632, 33, 34, 35syl3anc 1368 . . . . . 6 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}))
37 simprrr 780 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦))
388ntrss2 22981 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 βŠ† βˆͺ 𝐾) β†’ ((intβ€˜πΎ)β€˜π‘¦) βŠ† 𝑦)
397, 10, 38syl2anc 582 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ ((intβ€˜πΎ)β€˜π‘¦) βŠ† 𝑦)
4039adantr 479 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ ((intβ€˜πΎ)β€˜π‘¦) βŠ† 𝑦)
4137, 40sstrd 3992 . . . . . 6 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† ((intβ€˜πΎ)β€˜π‘¦)))) β†’ (𝐹 β€œ π‘₯) βŠ† 𝑦)
4228, 36, 41reximssdv 3170 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦)
4342ralrimiva 3143 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦)
443, 43jca 510 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦))
4544ex 411 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦)))
46 simpll2 1210 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4746, 6syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ 𝐾 ∈ Top)
48 simprl 769 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ 𝑦 ∈ 𝐾)
49 simprr 771 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ (πΉβ€˜π‘ƒ) ∈ 𝑦)
50 opnneip 23043 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}))
5147, 48, 49, 50syl3anc 1368 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ 𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}))
52 simpl1 1188 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5352ad2antrr 724 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5453, 31syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝐽 ∈ Top)
55 simprl 769 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}))
56 eqid 2728 . . . . . . . . . . . . . 14 βˆͺ 𝐽 = βˆͺ 𝐽
5756neii1 23030 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ π‘₯ βŠ† βˆͺ 𝐽)
5854, 55, 57syl2anc 582 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ π‘₯ βŠ† βˆͺ 𝐽)
5956ntropn 22973 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ π‘₯ βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
6054, 58, 59syl2anc 582 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
61 simpll3 1211 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ 𝑃 ∈ 𝑋)
6261adantr 479 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝑃 ∈ 𝑋)
63 toponuni 22836 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
6453, 63syl 17 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝑋 = βˆͺ 𝐽)
6562, 64eleqtrd 2831 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝑃 ∈ βˆͺ 𝐽)
6665snssd 4817 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ {𝑃} βŠ† βˆͺ 𝐽)
6756neiint 23028 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ {𝑃} βŠ† βˆͺ 𝐽 ∧ π‘₯ βŠ† βˆͺ 𝐽) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ {𝑃} βŠ† ((intβ€˜π½)β€˜π‘₯)))
6854, 66, 58, 67syl3anc 1368 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ {𝑃} βŠ† ((intβ€˜π½)β€˜π‘₯)))
6955, 68mpbid 231 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ {𝑃} βŠ† ((intβ€˜π½)β€˜π‘₯))
70 snssg 4792 . . . . . . . . . . . . 13 (𝑃 ∈ 𝑋 β†’ (𝑃 ∈ ((intβ€˜π½)β€˜π‘₯) ↔ {𝑃} βŠ† ((intβ€˜π½)β€˜π‘₯)))
7162, 70syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ (𝑃 ∈ ((intβ€˜π½)β€˜π‘₯) ↔ {𝑃} βŠ† ((intβ€˜π½)β€˜π‘₯)))
7269, 71mpbird 256 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ 𝑃 ∈ ((intβ€˜π½)β€˜π‘₯))
7356ntrss2 22981 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ π‘₯ βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
7454, 58, 73syl2anc 582 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
75 imass2 6111 . . . . . . . . . . . . 13 (((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯ β†’ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† (𝐹 β€œ π‘₯))
7674, 75syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† (𝐹 β€œ π‘₯))
77 simprr 771 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ (𝐹 β€œ π‘₯) βŠ† 𝑦)
7876, 77sstrd 3992 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† 𝑦)
79 eleq2 2818 . . . . . . . . . . . . 13 (𝑧 = ((intβ€˜π½)β€˜π‘₯) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ ((intβ€˜π½)β€˜π‘₯)))
80 imaeq2 6064 . . . . . . . . . . . . . 14 (𝑧 = ((intβ€˜π½)β€˜π‘₯) β†’ (𝐹 β€œ 𝑧) = (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)))
8180sseq1d 4013 . . . . . . . . . . . . 13 (𝑧 = ((intβ€˜π½)β€˜π‘₯) β†’ ((𝐹 β€œ 𝑧) βŠ† 𝑦 ↔ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† 𝑦))
8279, 81anbi12d 630 . . . . . . . . . . . 12 (𝑧 = ((intβ€˜π½)β€˜π‘₯) β†’ ((𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦) ↔ (𝑃 ∈ ((intβ€˜π½)β€˜π‘₯) ∧ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† 𝑦)))
8382rspcev 3611 . . . . . . . . . . 11 ((((intβ€˜π½)β€˜π‘₯) ∈ 𝐽 ∧ (𝑃 ∈ ((intβ€˜π½)β€˜π‘₯) ∧ (𝐹 β€œ ((intβ€˜π½)β€˜π‘₯)) βŠ† 𝑦)) β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦))
8460, 72, 78, 83syl12anc 835 . . . . . . . . . 10 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) ∧ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦))
8584rexlimdvaa 3153 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦)))
8651, 85embantd 59 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ ((𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦)))
8786ex 411 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ ((𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦))))
8887com23 86 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ ((𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦))))
8988exp4a 430 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((𝑦 ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)}) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ (𝑦 ∈ 𝐾 β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦)))))
9089ralimdv2 3160 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦))))
9190imdistanda 570 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦)))))
92 iscnp 23161 . . 3 ((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 ∈ 𝑧 ∧ (𝐹 β€œ 𝑧) βŠ† 𝑦)))))
9391, 92sylibrd 258 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)))
9445, 93impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ((neiβ€˜πΎ)β€˜{(πΉβ€˜π‘ƒ)})βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝐹 β€œ π‘₯) βŠ† 𝑦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067   βŠ† wss 3949  {csn 4632  βˆͺ cuni 4912   β€œ cima 5685  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  Topctop 22815  TopOnctopon 22832  intcnt 22941  neicnei 23021   CnP ccnp 23149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-top 22816  df-topon 22833  df-ntr 22944  df-nei 23022  df-cnp 23152
This theorem is referenced by:  cnnei  23206
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