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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  istopclsd Structured version   Visualization version   GIF version
Theorem istopclsd 41520
Description: A closure function which satisfies sscls 22567, clsidm 22578, cls0 22591, and clsun 35299 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
istopclsd.b (πœ‘ β†’ 𝐡 ∈ 𝑉)
istopclsd.f (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)
istopclsd.e ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))
istopclsd.i ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))
istopclsd.z (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
istopclsd.u ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† 𝐡) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)))
istopclsd.j 𝐽 = {𝑧 ∈ 𝒫 𝐡 ∣ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)}
Assertion
Ref Expression
istopclsd (πœ‘ β†’ (𝐽 ∈ (TopOnβ€˜π΅) ∧ (clsβ€˜π½) = 𝐹))
Distinct variable groups:   π‘₯,𝐡,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧   π‘₯,𝐹,𝑦,𝑧   π‘₯,𝐽,𝑦   π‘₯,𝑉,𝑦,𝑧
Allowed substitution hint:   𝐽(𝑧)
Proof of Theorem istopclsd
StepHypRef Expression
1 istopclsd.j . . . 4 𝐽 = {𝑧 ∈ 𝒫 𝐡 ∣ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)}
2 istopclsd.f . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)
32ffnd 6718 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn 𝒫 𝐡)
43adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ 𝐹 Fn 𝒫 𝐡)
5 difss 4131 . . . . . . . . 9 (𝐡 βˆ– 𝑧) βŠ† 𝐡
6 istopclsd.b . . . . . . . . . 10 (πœ‘ β†’ 𝐡 ∈ 𝑉)
7 elpw2g 5344 . . . . . . . . . 10 (𝐡 ∈ 𝑉 β†’ ((𝐡 βˆ– 𝑧) ∈ 𝒫 𝐡 ↔ (𝐡 βˆ– 𝑧) βŠ† 𝐡))
86, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ ((𝐡 βˆ– 𝑧) ∈ 𝒫 𝐡 ↔ (𝐡 βˆ– 𝑧) βŠ† 𝐡))
95, 8mpbiri 257 . . . . . . . 8 (πœ‘ β†’ (𝐡 βˆ– 𝑧) ∈ 𝒫 𝐡)
109adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (𝐡 βˆ– 𝑧) ∈ 𝒫 𝐡)
11 fnelfp 7175 . . . . . . 7 ((𝐹 Fn 𝒫 𝐡 ∧ (𝐡 βˆ– 𝑧) ∈ 𝒫 𝐡) β†’ ((𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)))
124, 10, 11syl2anc 584 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)))
1312bicomd 222 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧) ↔ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )))
1413rabbidva 3439 . . . 4 (πœ‘ β†’ {𝑧 ∈ 𝒫 𝐡 ∣ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)} = {𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )})
151, 14eqtrid 2784 . . 3 (πœ‘ β†’ 𝐽 = {𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )})
16 istopclsd.e . . . . . 6 ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))
17 simp1 1136 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ πœ‘)
18 simp2 1137 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ π‘₯ βŠ† 𝐡)
19 simp3 1138 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ 𝑦 βŠ† π‘₯)
2019, 18sstrd 3992 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ 𝑦 βŠ† 𝐡)
21 istopclsd.u . . . . . . . . 9 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† 𝐡) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)))
2217, 18, 20, 21syl3anc 1371 . . . . . . . 8 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)))
23 ssequn2 4183 . . . . . . . . . . 11 (𝑦 βŠ† π‘₯ ↔ (π‘₯ βˆͺ 𝑦) = π‘₯)
2423biimpi 215 . . . . . . . . . 10 (𝑦 βŠ† π‘₯ β†’ (π‘₯ βˆͺ 𝑦) = π‘₯)
25243ad2ant3 1135 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (π‘₯ βˆͺ 𝑦) = π‘₯)
2625fveq2d 6895 . . . . . . . 8 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = (πΉβ€˜π‘₯))
2722, 26eqtr3d 2774 . . . . . . 7 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)) = (πΉβ€˜π‘₯))
28 ssequn2 4183 . . . . . . 7 ((πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯) ↔ ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)) = (πΉβ€˜π‘₯))
2927, 28sylibr 233 . . . . . 6 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯))
30 istopclsd.i . . . . . 6 ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))
316, 2, 16, 29, 30ismrcd1 41518 . . . . 5 (πœ‘ β†’ dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅))
32 istopclsd.z . . . . . 6 (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
33 0elpw 5354 . . . . . . 7 βˆ… ∈ 𝒫 𝐡
34 fnelfp 7175 . . . . . . 7 ((𝐹 Fn 𝒫 𝐡 ∧ βˆ… ∈ 𝒫 𝐡) β†’ (βˆ… ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜βˆ…) = βˆ…))
353, 33, 34sylancl 586 . . . . . 6 (πœ‘ β†’ (βˆ… ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜βˆ…) = βˆ…))
3632, 35mpbird 256 . . . . 5 (πœ‘ β†’ βˆ… ∈ dom (𝐹 ∩ I ))
37 simp1 1136 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ πœ‘)
38 inss1 4228 . . . . . . . . . . . . 13 (𝐹 ∩ I ) βŠ† 𝐹
39 dmss 5902 . . . . . . . . . . . . 13 ((𝐹 ∩ I ) βŠ† 𝐹 β†’ dom (𝐹 ∩ I ) βŠ† dom 𝐹)
4038, 39ax-mp 5 . . . . . . . . . . . 12 dom (𝐹 ∩ I ) βŠ† dom 𝐹
4140, 2fssdm 6737 . . . . . . . . . . 11 (πœ‘ β†’ dom (𝐹 ∩ I ) βŠ† 𝒫 𝐡)
42413ad2ant1 1133 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ dom (𝐹 ∩ I ) βŠ† 𝒫 𝐡)
43 simp2 1137 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ π‘₯ ∈ dom (𝐹 ∩ I ))
4442, 43sseldd 3983 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ π‘₯ ∈ 𝒫 𝐡)
4544elpwid 4611 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ π‘₯ βŠ† 𝐡)
46 simp3 1138 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ 𝑦 ∈ dom (𝐹 ∩ I ))
4742, 46sseldd 3983 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ 𝑦 ∈ 𝒫 𝐡)
4847elpwid 4611 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ 𝑦 βŠ† 𝐡)
4937, 45, 48, 21syl3anc 1371 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)))
5033ad2ant1 1133 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ 𝐹 Fn 𝒫 𝐡)
51 fnelfp 7175 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐡 ∧ π‘₯ ∈ 𝒫 𝐡) β†’ (π‘₯ ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜π‘₯) = π‘₯))
5250, 44, 51syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (π‘₯ ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜π‘₯) = π‘₯))
5343, 52mpbid 231 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (πΉβ€˜π‘₯) = π‘₯)
54 fnelfp 7175 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐡 ∧ 𝑦 ∈ 𝒫 𝐡) β†’ (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜π‘¦) = 𝑦))
5550, 47, 54syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜π‘¦) = 𝑦))
5646, 55mpbid 231 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (πΉβ€˜π‘¦) = 𝑦)
5753, 56uneq12d 4164 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)) = (π‘₯ βˆͺ 𝑦))
5849, 57eqtrd 2772 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = (π‘₯ βˆͺ 𝑦))
5945, 48unssd 4186 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (π‘₯ βˆͺ 𝑦) βŠ† 𝐡)
60 vex 3478 . . . . . . . . . 10 π‘₯ ∈ V
61 vex 3478 . . . . . . . . . 10 𝑦 ∈ V
6260, 61unex 7735 . . . . . . . . 9 (π‘₯ βˆͺ 𝑦) ∈ V
6362elpw 4606 . . . . . . . 8 ((π‘₯ βˆͺ 𝑦) ∈ 𝒫 𝐡 ↔ (π‘₯ βˆͺ 𝑦) βŠ† 𝐡)
6459, 63sylibr 233 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (π‘₯ βˆͺ 𝑦) ∈ 𝒫 𝐡)
65 fnelfp 7175 . . . . . . 7 ((𝐹 Fn 𝒫 𝐡 ∧ (π‘₯ βˆͺ 𝑦) ∈ 𝒫 𝐡) β†’ ((π‘₯ βˆͺ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = (π‘₯ βˆͺ 𝑦)))
6650, 64, 65syl2anc 584 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ ((π‘₯ βˆͺ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = (π‘₯ βˆͺ 𝑦)))
6758, 66mpbird 256 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) β†’ (π‘₯ βˆͺ 𝑦) ∈ dom (𝐹 ∩ I ))
68 eqid 2732 . . . . 5 {𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )}
6931, 36, 67, 68mretopd 22603 . . . 4 (πœ‘ β†’ ({𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOnβ€˜π΅) ∧ dom (𝐹 ∩ I ) = (Clsdβ€˜{𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )})))
7069simpld 495 . . 3 (πœ‘ β†’ {𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOnβ€˜π΅))
7115, 70eqeltrd 2833 . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π΅))
72 topontop 22422 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐽 ∈ Top)
7371, 72syl 17 . . . . 5 (πœ‘ β†’ 𝐽 ∈ Top)
74 eqid 2732 . . . . . 6 (mrClsβ€˜(Clsdβ€˜π½)) = (mrClsβ€˜(Clsdβ€˜π½))
7574mrccls 22590 . . . . 5 (𝐽 ∈ Top β†’ (clsβ€˜π½) = (mrClsβ€˜(Clsdβ€˜π½)))
7673, 75syl 17 . . . 4 (πœ‘ β†’ (clsβ€˜π½) = (mrClsβ€˜(Clsdβ€˜π½)))
7769simprd 496 . . . . . 6 (πœ‘ β†’ dom (𝐹 ∩ I ) = (Clsdβ€˜{𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )}))
7815fveq2d 6895 . . . . . 6 (πœ‘ β†’ (Clsdβ€˜π½) = (Clsdβ€˜{𝑧 ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– 𝑧) ∈ dom (𝐹 ∩ I )}))
7977, 78eqtr4d 2775 . . . . 5 (πœ‘ β†’ dom (𝐹 ∩ I ) = (Clsdβ€˜π½))
8079fveq2d 6895 . . . 4 (πœ‘ β†’ (mrClsβ€˜dom (𝐹 ∩ I )) = (mrClsβ€˜(Clsdβ€˜π½)))
8176, 80eqtr4d 2775 . . 3 (πœ‘ β†’ (clsβ€˜π½) = (mrClsβ€˜dom (𝐹 ∩ I )))
826, 2, 16, 29, 30ismrcd2 41519 . . 3 (πœ‘ β†’ 𝐹 = (mrClsβ€˜dom (𝐹 ∩ I )))
8381, 82eqtr4d 2775 . 2 (πœ‘ β†’ (clsβ€˜π½) = 𝐹)
8471, 83jca 512 1 (πœ‘ β†’ (𝐽 ∈ (TopOnβ€˜π΅) ∧ (clsβ€˜π½) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602   I cid 5573  dom cdm 5676   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  mrClscmrc 17529  Topctop 22402  TopOnctopon 22419  Clsdccld 22527  clsccl 22529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-mre 17532  df-mrc 17533  df-top 22403  df-topon 22420  df-cld 22530  df-cls 22532
This theorem is referenced by: (None)
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