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Theorem kqt0lem 23240
Description: Lemma for kqt0 23250. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqt0lem (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ Kol2)
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem kqt0lem
Dummy variables 𝑀 𝑧 π‘Ž 𝑏 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . . . . 10 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqopn 23238 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) β†’ (𝐹 β€œ 𝑀) ∈ (KQβ€˜π½))
32adantlr 714 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑀 ∈ 𝐽) β†’ (𝐹 β€œ 𝑀) ∈ (KQβ€˜π½))
4 eleq2 2823 . . . . . . . . . 10 (𝑧 = (𝐹 β€œ 𝑀) β†’ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
5 eleq2 2823 . . . . . . . . . 10 (𝑧 = (𝐹 β€œ 𝑀) β†’ ((πΉβ€˜π‘) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
64, 5bibi12d 346 . . . . . . . . 9 (𝑧 = (𝐹 β€œ 𝑀) β†’ (((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) ↔ ((πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀) ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀))))
76rspcv 3609 . . . . . . . 8 ((𝐹 β€œ 𝑀) ∈ (KQβ€˜π½) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ ((πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀) ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀))))
83, 7syl 17 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑀 ∈ 𝐽) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ ((πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀) ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀))))
91kqfvima 23234 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽 ∧ π‘Ž ∈ 𝑋) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
1093expa 1119 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ π‘Ž ∈ 𝑋) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
1110adantrr 716 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
121kqfvima 23234 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
13123expa 1119 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
1413adantrl 715 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
1511, 14bibi12d 346 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ((π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀) ↔ ((πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀) ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀))))
1615an32s 651 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑀 ∈ 𝐽) β†’ ((π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀) ↔ ((πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀) ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀))))
178, 16sylibrd 259 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑀 ∈ 𝐽) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀)))
1817ralrimdva 3155 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ βˆ€π‘€ ∈ 𝐽 (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀)))
191kqfeq 23228 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ↔ βˆ€π‘¦ ∈ 𝐽 (π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20193expb 1121 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ↔ βˆ€π‘¦ ∈ 𝐽 (π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21 elequ2 2122 . . . . . . . 8 (𝑦 = 𝑀 β†’ (π‘Ž ∈ 𝑦 ↔ π‘Ž ∈ 𝑀))
22 elequ2 2122 . . . . . . . 8 (𝑦 = 𝑀 β†’ (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑀))
2321, 22bibi12d 346 . . . . . . 7 (𝑦 = 𝑀 β†’ ((π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀)))
2423cbvralvw 3235 . . . . . 6 (βˆ€π‘¦ ∈ 𝐽 (π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ βˆ€π‘€ ∈ 𝐽 (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀))
2520, 24bitrdi 287 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ↔ βˆ€π‘€ ∈ 𝐽 (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀)))
2618, 25sylibrd 259 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘)))
2726ralrimivva 3201 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘)))
281kqffn 23229 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
29 eleq1 2822 . . . . . . . . . 10 (𝑒 = (πΉβ€˜π‘Ž) β†’ (𝑒 ∈ 𝑧 ↔ (πΉβ€˜π‘Ž) ∈ 𝑧))
3029bibi1d 344 . . . . . . . . 9 (𝑒 = (πΉβ€˜π‘Ž) β†’ ((𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
3130ralbidv 3178 . . . . . . . 8 (𝑒 = (πΉβ€˜π‘Ž) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32 eqeq1 2737 . . . . . . . 8 (𝑒 = (πΉβ€˜π‘Ž) β†’ (𝑒 = 𝑣 ↔ (πΉβ€˜π‘Ž) = 𝑣))
3331, 32imbi12d 345 . . . . . . 7 (𝑒 = (πΉβ€˜π‘Ž) β†’ ((βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣) ↔ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣)))
3433ralbidv 3178 . . . . . 6 (𝑒 = (πΉβ€˜π‘Ž) β†’ (βˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣) ↔ βˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣)))
3534ralrn 7090 . . . . 5 (𝐹 Fn 𝑋 β†’ (βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣) ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣)))
36 eleq1 2822 . . . . . . . . . 10 (𝑣 = (πΉβ€˜π‘) β†’ (𝑣 ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧))
3736bibi2d 343 . . . . . . . . 9 (𝑣 = (πΉβ€˜π‘) β†’ (((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧)))
3837ralbidv 3178 . . . . . . . 8 (𝑣 = (πΉβ€˜π‘) β†’ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧)))
39 eqeq2 2745 . . . . . . . 8 (𝑣 = (πΉβ€˜π‘) β†’ ((πΉβ€˜π‘Ž) = 𝑣 ↔ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘)))
4038, 39imbi12d 345 . . . . . . 7 (𝑣 = (πΉβ€˜π‘) β†’ ((βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣) ↔ (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))))
4140ralrn 7090 . . . . . 6 (𝐹 Fn 𝑋 β†’ (βˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣) ↔ βˆ€π‘ ∈ 𝑋 (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))))
4241ralbidv 3178 . . . . 5 (𝐹 Fn 𝑋 β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = 𝑣) ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))))
4335, 42bitrd 279 . . . 4 (𝐹 Fn 𝑋 β†’ (βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣) ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))))
4428, 43syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣) ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (βˆ€π‘§ ∈ (KQβ€˜π½)((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))))
4527, 44mpbird 257 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣))
461kqtopon 23231 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
47 ist0-2 22848 . . 3 ((KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹) β†’ ((KQβ€˜π½) ∈ Kol2 ↔ βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣)))
4846, 47syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((KQβ€˜π½) ∈ Kol2 ↔ βˆ€π‘’ ∈ ran πΉβˆ€π‘£ ∈ ran 𝐹(βˆ€π‘§ ∈ (KQβ€˜π½)(𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) β†’ 𝑒 = 𝑣)))
4945, 48mpbird 257 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   ↦ cmpt 5232  ran crn 5678   β€œ cima 5680   Fn wfn 6539  β€˜cfv 6544  TopOnctopon 22412  Kol2ct0 22810  KQckq 23197
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-rep 5286  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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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-iun 5000  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-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453  df-top 22396  df-topon 22413  df-t0 22817  df-kq 23198
This theorem is referenced by:  kqt0  23250  t0kq  23322
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