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Theorem iscncl 22993
Description: A characterization of a continuity function using closed sets. Theorem 1(d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscncl ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,π‘Œ

Proof of Theorem iscncl
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 cnf2 22973 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1116 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3 cnclima 22992 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
43ralrimiva 3144 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
54adantl 480 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
62, 5jca 510 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½)))
7 simprl 767 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
8 toponuni 22636 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
98ad3antrrr 726 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ 𝑋 = βˆͺ 𝐽)
10 simplrl 773 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
11 fimacnv 6738 . . . . . . . . . . 11 (𝐹:π‘‹βŸΆπ‘Œ β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
1211eqcomd 2736 . . . . . . . . . 10 (𝐹:π‘‹βŸΆπ‘Œ β†’ 𝑋 = (◑𝐹 β€œ π‘Œ))
1310, 12syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ 𝑋 = (◑𝐹 β€œ π‘Œ))
149, 13eqtr3d 2772 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ βˆͺ 𝐽 = (◑𝐹 β€œ π‘Œ))
1514difeq1d 4120 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ π‘₯)) = ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ π‘₯)))
16 ffun 6719 . . . . . . . 8 (𝐹:π‘‹βŸΆπ‘Œ β†’ Fun 𝐹)
17 funcnvcnv 6614 . . . . . . . 8 (Fun 𝐹 β†’ Fun ◑◑𝐹)
18 imadif 6631 . . . . . . . 8 (Fun ◑◑𝐹 β†’ (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)) = ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ π‘₯)))
1910, 16, 17, 184syl 19 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)) = ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ π‘₯)))
2015, 19eqtr4d 2773 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)))
21 imaeq2 6054 . . . . . . . 8 (𝑦 = (π‘Œ βˆ– π‘₯) β†’ (◑𝐹 β€œ 𝑦) = (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)))
2221eleq1d 2816 . . . . . . 7 (𝑦 = (π‘Œ βˆ– π‘₯) β†’ ((◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½) ↔ (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)) ∈ (Clsdβ€˜π½)))
23 simplrr 774 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
24 toponuni 22636 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
2524ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ π‘Œ = βˆͺ 𝐾)
2625difeq1d 4120 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (π‘Œ βˆ– π‘₯) = (βˆͺ 𝐾 βˆ– π‘₯))
27 topontop 22635 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2827ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ 𝐾 ∈ Top)
29 eqid 2730 . . . . . . . . . 10 βˆͺ 𝐾 = βˆͺ 𝐾
3029opncld 22757 . . . . . . . . 9 ((𝐾 ∈ Top ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐾 βˆ– π‘₯) ∈ (Clsdβ€˜πΎ))
3128, 30sylancom 586 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐾 βˆ– π‘₯) ∈ (Clsdβ€˜πΎ))
3226, 31eqeltrd 2831 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (π‘Œ βˆ– π‘₯) ∈ (Clsdβ€˜πΎ))
3322, 23, 32rspcdva 3612 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)) ∈ (Clsdβ€˜π½))
3420, 33eqeltrd 2831 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ π‘₯)) ∈ (Clsdβ€˜π½))
35 topontop 22635 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3635ad3antrrr 726 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ 𝐽 ∈ Top)
37 cnvimass 6079 . . . . . . . 8 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
3837, 10fssdm 6736 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) βŠ† 𝑋)
3938, 9sseqtrd 4021 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽)
40 eqid 2730 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
4140isopn2 22756 . . . . . 6 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽) β†’ ((◑𝐹 β€œ π‘₯) ∈ 𝐽 ↔ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ π‘₯)) ∈ (Clsdβ€˜π½)))
4236, 39, 41syl2anc 582 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ π‘₯) ∈ 𝐽 ↔ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ π‘₯)) ∈ (Clsdβ€˜π½)))
4334, 42mpbird 256 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
4443ralrimiva 3144 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) β†’ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)
45 iscn 22959 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
4645adantr 479 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
477, 44, 46mpbir2and 709 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
486, 47impbida 797 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βˆ– cdif 3944   βŠ† wss 3947  βˆͺ cuni 4907  β—‘ccnv 5674   β€œ cima 5678  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Topctop 22615  TopOnctopon 22632  Clsdccld 22740   Cn ccn 22948
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-map 8824  df-top 22616  df-topon 22633  df-cld 22743  df-cn 22951
This theorem is referenced by:  cncls2  22997  paste  23018  cmphaushmeo  23524  ubthlem1  30390  ubthlem2  30391  rhmpreimacn  33163
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