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Theorem iscncl 22773
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 22753 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1119 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3 cnclima 22772 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
43ralrimiva 3147 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
54adantl 483 . . 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 513 . 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 770 . . 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 22416 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
98ad3antrrr 729 . . . . . . . . 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 776 . . . . . . . . . 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 6740 . . . . . . . . . . 11 (𝐹:π‘‹βŸΆπ‘Œ β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
1211eqcomd 2739 . . . . . . . . . 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 2775 . . . . . . . 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 4122 . . . . . . 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 6721 . . . . . . . 8 (𝐹:π‘‹βŸΆπ‘Œ β†’ Fun 𝐹)
17 funcnvcnv 6616 . . . . . . . 8 (Fun 𝐹 β†’ Fun ◑◑𝐹)
18 imadif 6633 . . . . . . . 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 2776 . . . . . 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 6056 . . . . . . . 8 (𝑦 = (π‘Œ βˆ– π‘₯) β†’ (◑𝐹 β€œ 𝑦) = (◑𝐹 β€œ (π‘Œ βˆ– π‘₯)))
2221eleq1d 2819 . . . . . . 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 777 . . . . . . 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 22416 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
2524ad3antlr 730 . . . . . . . . 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 4122 . . . . . . . 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 22415 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2827ad3antlr 730 . . . . . . . . 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 2733 . . . . . . . . . 10 βˆͺ 𝐾 = βˆͺ 𝐾
3029opncld 22537 . . . . . . . . 9 ((𝐾 ∈ Top ∧ π‘₯ ∈ 𝐾) β†’ (βˆͺ 𝐾 βˆ– π‘₯) ∈ (Clsdβ€˜πΎ))
3128, 30sylancom 589 . . . . . . . 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 2834 . . . . . . 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 3614 . . . . . 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 2834 . . . . 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 22415 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3635ad3antrrr 729 . . . . . 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 6081 . . . . . . . 8 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
3837, 10fssdm 6738 . . . . . . 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 4023 . . . . . 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 2733 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
4140isopn2 22536 . . . . . 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 585 . . . . 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 257 . . . 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 3147 . . 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 22739 . . . 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 482 . . 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 712 . 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 800 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βˆ– cdif 3946   βŠ† wss 3949  βˆͺ cuni 4909  β—‘ccnv 5676   β€œ cima 5680  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  Clsdccld 22520   Cn ccn 22728
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-cld 22523  df-cn 22731
This theorem is referenced by:  cncls2  22777  paste  22798  cmphaushmeo  23304  ubthlem1  30123  ubthlem2  30124  rhmpreimacn  32865
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