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Theorem isr0 23088
Description: The property "𝐽 is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains π‘₯ also contains 𝑦, so there is no separation, then π‘₯ and 𝑦 are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
isr0 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((KQβ€˜π½) ∈ Fre ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ))))
Distinct variable groups:   𝑀,π‘œ,π‘₯,𝑦,𝑧,𝐽   π‘œ,𝐹,𝑀,𝑧   π‘œ,𝑋,𝑀,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem isr0
Dummy variables π‘Ž 𝑏 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . . . . . . 12 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqid 23079 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 ∈ (𝐽 Cn (KQβ€˜π½)))
32ad2antrr 724 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ 𝐹 ∈ (𝐽 Cn (KQβ€˜π½)))
4 cnima 22616 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn (KQβ€˜π½)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ (◑𝐹 β€œ 𝑣) ∈ 𝐽)
53, 4sylan 580 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ (◑𝐹 β€œ 𝑣) ∈ 𝐽)
6 eleq2 2826 . . . . . . . . . . 11 (π‘œ = (◑𝐹 β€œ 𝑣) β†’ (𝑧 ∈ π‘œ ↔ 𝑧 ∈ (◑𝐹 β€œ 𝑣)))
7 eleq2 2826 . . . . . . . . . . 11 (π‘œ = (◑𝐹 β€œ 𝑣) β†’ (𝑀 ∈ π‘œ ↔ 𝑀 ∈ (◑𝐹 β€œ 𝑣)))
86, 7imbi12d 344 . . . . . . . . . 10 (π‘œ = (◑𝐹 β€œ 𝑣) β†’ ((𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) ↔ (𝑧 ∈ (◑𝐹 β€œ 𝑣) β†’ 𝑀 ∈ (◑𝐹 β€œ 𝑣))))
98rspcv 3577 . . . . . . . . 9 ((◑𝐹 β€œ 𝑣) ∈ 𝐽 β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ (𝑧 ∈ (◑𝐹 β€œ 𝑣) β†’ 𝑀 ∈ (◑𝐹 β€œ 𝑣))))
105, 9syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ (𝑧 ∈ (◑𝐹 β€œ 𝑣) β†’ 𝑀 ∈ (◑𝐹 β€œ 𝑣))))
111kqffn 23076 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
1211ad2antrr 724 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ 𝐹 Fn 𝑋)
1312adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ 𝐹 Fn 𝑋)
14 fnfun 6602 . . . . . . . . . . 11 (𝐹 Fn 𝑋 β†’ Fun 𝐹)
1513, 14syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ Fun 𝐹)
16 simprl 769 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ 𝑧 ∈ 𝑋)
1716adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ 𝑧 ∈ 𝑋)
1813fndmd 6607 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ dom 𝐹 = 𝑋)
1917, 18eleqtrrd 2840 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ 𝑧 ∈ dom 𝐹)
20 fvimacnv 7003 . . . . . . . . . 10 ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) β†’ ((πΉβ€˜π‘§) ∈ 𝑣 ↔ 𝑧 ∈ (◑𝐹 β€œ 𝑣)))
2115, 19, 20syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ ((πΉβ€˜π‘§) ∈ 𝑣 ↔ 𝑧 ∈ (◑𝐹 β€œ 𝑣)))
22 simprr 771 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ 𝑀 ∈ 𝑋)
2322adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ 𝑀 ∈ 𝑋)
2423, 18eleqtrrd 2840 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ 𝑀 ∈ dom 𝐹)
25 fvimacnv 7003 . . . . . . . . . 10 ((Fun 𝐹 ∧ 𝑀 ∈ dom 𝐹) β†’ ((πΉβ€˜π‘€) ∈ 𝑣 ↔ 𝑀 ∈ (◑𝐹 β€œ 𝑣)))
2615, 24, 25syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ ((πΉβ€˜π‘€) ∈ 𝑣 ↔ 𝑀 ∈ (◑𝐹 β€œ 𝑣)))
2721, 26imbi12d 344 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ (((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) ↔ (𝑧 ∈ (◑𝐹 β€œ 𝑣) β†’ 𝑀 ∈ (◑𝐹 β€œ 𝑣))))
2810, 27sylibrd 258 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) ∧ 𝑣 ∈ (KQβ€˜π½)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ ((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣)))
2928ralrimdva 3151 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣)))
30 simplr 767 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (KQβ€˜π½) ∈ Fre)
31 fnfvelrn 7031 . . . . . . . . 9 ((𝐹 Fn 𝑋 ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) ∈ ran 𝐹)
3212, 16, 31syl2anc 584 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (πΉβ€˜π‘§) ∈ ran 𝐹)
331kqtopon 23078 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
3433ad2antrr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
35 toponuni 22263 . . . . . . . . 9 ((KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹) β†’ ran 𝐹 = βˆͺ (KQβ€˜π½))
3634, 35syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ ran 𝐹 = βˆͺ (KQβ€˜π½))
3732, 36eleqtrd 2839 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (πΉβ€˜π‘§) ∈ βˆͺ (KQβ€˜π½))
38 fnfvelrn 7031 . . . . . . . . 9 ((𝐹 Fn 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ (πΉβ€˜π‘€) ∈ ran 𝐹)
3912, 22, 38syl2anc 584 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (πΉβ€˜π‘€) ∈ ran 𝐹)
4039, 36eleqtrd 2839 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (πΉβ€˜π‘€) ∈ βˆͺ (KQβ€˜π½))
41 eqid 2736 . . . . . . . 8 βˆͺ (KQβ€˜π½) = βˆͺ (KQβ€˜π½)
4241t1sep2 22720 . . . . . . 7 (((KQβ€˜π½) ∈ Fre ∧ (πΉβ€˜π‘§) ∈ βˆͺ (KQβ€˜π½) ∧ (πΉβ€˜π‘€) ∈ βˆͺ (KQβ€˜π½)) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€)))
4330, 37, 40, 42syl3anc 1371 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€)))
4429, 43syld 47 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€)))
451kqfeq 23075 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦)))
46 eleq2 2826 . . . . . . . . . 10 (π‘œ = 𝑦 β†’ (𝑧 ∈ π‘œ ↔ 𝑧 ∈ 𝑦))
47 eleq2 2826 . . . . . . . . . 10 (π‘œ = 𝑦 β†’ (𝑀 ∈ π‘œ ↔ 𝑀 ∈ 𝑦))
4846, 47bibi12d 345 . . . . . . . . 9 (π‘œ = 𝑦 β†’ ((𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ) ↔ (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦)))
4948cbvralvw 3225 . . . . . . . 8 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦))
5045, 49bitr4di 288 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
51503expb 1120 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
5251adantlr 713 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
5344, 52sylibd 238 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
5453ralrimivva 3197 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre) β†’ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
5554ex 413 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((KQβ€˜π½) ∈ Fre β†’ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ))))
561kqopn 23085 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘œ ∈ 𝐽) β†’ (𝐹 β€œ π‘œ) ∈ (KQβ€˜π½))
5756ad4ant14 750 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝐹 β€œ π‘œ) ∈ (KQβ€˜π½))
58 eleq2 2826 . . . . . . . . . . . 12 (𝑣 = (𝐹 β€œ π‘œ) β†’ ((πΉβ€˜π‘§) ∈ 𝑣 ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ)))
59 eleq2 2826 . . . . . . . . . . . 12 (𝑣 = (𝐹 β€œ π‘œ) β†’ ((πΉβ€˜π‘€) ∈ 𝑣 ↔ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ)))
6058, 59imbi12d 344 . . . . . . . . . . 11 (𝑣 = (𝐹 β€œ π‘œ) β†’ (((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) ↔ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ) β†’ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ))))
6160rspcv 3577 . . . . . . . . . 10 ((𝐹 β€œ π‘œ) ∈ (KQβ€˜π½) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ) β†’ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ))))
6257, 61syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ) β†’ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ))))
631kqfvima 23081 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘œ ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ π‘œ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ)))
64633expa 1118 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘œ ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ π‘œ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ)))
6564an32s 650 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝑧 ∈ π‘œ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ)))
6665adantlr 713 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝑧 ∈ π‘œ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ)))
671kqfvima 23081 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘œ ∈ 𝐽 ∧ 𝑀 ∈ 𝑋) β†’ (𝑀 ∈ π‘œ ↔ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ)))
68673expa 1118 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘œ ∈ 𝐽) ∧ 𝑀 ∈ 𝑋) β†’ (𝑀 ∈ π‘œ ↔ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ)))
6968an32s 650 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝑀 ∈ π‘œ ↔ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ)))
7069adantllr 717 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝑀 ∈ π‘œ ↔ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ)))
7166, 70imbi12d 344 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ ((𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) ↔ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘œ) β†’ (πΉβ€˜π‘€) ∈ (𝐹 β€œ π‘œ))))
7262, 71sylibrd 258 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ)))
7372ralrimdva 3151 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ)))
741kqfval 23074 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
7574adantr 481 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ (πΉβ€˜π‘§) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
761kqfval 23074 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑋) β†’ (πΉβ€˜π‘€) = {𝑦 ∈ 𝐽 ∣ 𝑀 ∈ 𝑦})
7776adantlr 713 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ (πΉβ€˜π‘€) = {𝑦 ∈ 𝐽 ∣ 𝑀 ∈ 𝑦})
7875, 77eqeq12d 2752 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑀 ∈ 𝑦}))
79 rabbi 3432 . . . . . . . . . 10 (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑀 ∈ 𝑦})
8049, 79bitri 274 . . . . . . . . 9 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑀 ∈ 𝑦})
8178, 80bitr4di 288 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)))
8281biimprd 247 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€)))
8373, 82imim12d 81 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) ∧ 𝑀 ∈ 𝑋) β†’ ((βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
8483ralimdva 3164 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋) β†’ (βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)) β†’ βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
8584ralimdva 3164 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)) β†’ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
86 eleq1 2825 . . . . . . . . . . 11 (π‘Ž = (πΉβ€˜π‘§) β†’ (π‘Ž ∈ 𝑣 ↔ (πΉβ€˜π‘§) ∈ 𝑣))
8786imbi1d 341 . . . . . . . . . 10 (π‘Ž = (πΉβ€˜π‘§) β†’ ((π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) ↔ ((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣)))
8887ralbidv 3174 . . . . . . . . 9 (π‘Ž = (πΉβ€˜π‘§) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) ↔ βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣)))
89 eqeq1 2740 . . . . . . . . 9 (π‘Ž = (πΉβ€˜π‘§) β†’ (π‘Ž = 𝑏 ↔ (πΉβ€˜π‘§) = 𝑏))
9088, 89imbi12d 344 . . . . . . . 8 (π‘Ž = (πΉβ€˜π‘§) β†’ ((βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏) ↔ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏)))
9190ralbidv 3174 . . . . . . 7 (π‘Ž = (πΉβ€˜π‘§) β†’ (βˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏) ↔ βˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏)))
9291ralrn 7038 . . . . . 6 (𝐹 Fn 𝑋 β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏) ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏)))
93 eleq1 2825 . . . . . . . . . . 11 (𝑏 = (πΉβ€˜π‘€) β†’ (𝑏 ∈ 𝑣 ↔ (πΉβ€˜π‘€) ∈ 𝑣))
9493imbi2d 340 . . . . . . . . . 10 (𝑏 = (πΉβ€˜π‘€) β†’ (((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) ↔ ((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣)))
9594ralbidv 3174 . . . . . . . . 9 (𝑏 = (πΉβ€˜π‘€) β†’ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) ↔ βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣)))
96 eqeq2 2748 . . . . . . . . 9 (𝑏 = (πΉβ€˜π‘€) β†’ ((πΉβ€˜π‘§) = 𝑏 ↔ (πΉβ€˜π‘§) = (πΉβ€˜π‘€)))
9795, 96imbi12d 344 . . . . . . . 8 (𝑏 = (πΉβ€˜π‘€) β†’ ((βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏) ↔ (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
9897ralrn 7038 . . . . . . 7 (𝐹 Fn 𝑋 β†’ (βˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏) ↔ βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
9998ralbidv 3174 . . . . . 6 (𝐹 Fn 𝑋 β†’ (βˆ€π‘§ ∈ 𝑋 βˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ (πΉβ€˜π‘§) = 𝑏) ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
10092, 99bitrd 278 . . . . 5 (𝐹 Fn 𝑋 β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏) ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
10111, 100syl 17 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏) ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘£ ∈ (KQβ€˜π½)((πΉβ€˜π‘§) ∈ 𝑣 β†’ (πΉβ€˜π‘€) ∈ 𝑣) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘€))))
10285, 101sylibrd 258 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)) β†’ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏)))
103 ist1-2 22698 . . . 4 ((KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹) β†’ ((KQβ€˜π½) ∈ Fre ↔ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏)))
10433, 103syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((KQβ€˜π½) ∈ Fre ↔ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(βˆ€π‘£ ∈ (KQβ€˜π½)(π‘Ž ∈ 𝑣 β†’ 𝑏 ∈ 𝑣) β†’ π‘Ž = 𝑏)))
105102, 104sylibrd 258 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ)) β†’ (KQβ€˜π½) ∈ Fre))
10655, 105impbid 211 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((KQβ€˜π½) ∈ Fre ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ β†’ 𝑀 ∈ π‘œ) β†’ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝑀 ∈ π‘œ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3064  {crab 3407  βˆͺ cuni 4865   ↦ cmpt 5188  β—‘ccnv 5632  dom cdm 5633  ran crn 5634   β€œ cima 5636  Fun wfun 6490   Fn wfn 6491  β€˜cfv 6496  (class class class)co 7357  TopOnctopon 22259   Cn ccn 22575  Frect1 22658  KQckq 23044
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-topgen 17325  df-qtop 17389  df-top 22243  df-topon 22260  df-cld 22370  df-cn 22578  df-t1 22665  df-kq 23045
This theorem is referenced by:  r0sep  23099
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