MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqt0lem Structured version   Visualization version   GIF version

Theorem kqt0lem 23087
Description: Lemma for kqt0 23097. (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 23085 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) β†’ (𝐹 β€œ 𝑀) ∈ (KQβ€˜π½))
32adantlr 713 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑀 ∈ 𝐽) β†’ (𝐹 β€œ 𝑀) ∈ (KQβ€˜π½))
4 eleq2 2826 . . . . . . . . . 10 (𝑧 = (𝐹 β€œ 𝑀) β†’ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
5 eleq2 2826 . . . . . . . . . 10 (𝑧 = (𝐹 β€œ 𝑀) β†’ ((πΉβ€˜π‘) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
64, 5bibi12d 345 . . . . . . . . 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 3577 . . . . . . . 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 23081 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽 ∧ π‘Ž ∈ 𝑋) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
1093expa 1118 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ π‘Ž ∈ 𝑋) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
1110adantrr 715 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘Ž ∈ 𝑀 ↔ (πΉβ€˜π‘Ž) ∈ (𝐹 β€œ 𝑀)))
121kqfvima 23081 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
13123expa 1118 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
1413adantrl 714 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝐽) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (𝑏 ∈ 𝑀 ↔ (πΉβ€˜π‘) ∈ (𝐹 β€œ 𝑀)))
1511, 14bibi12d 345 . . . . . . . 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 650 . . . . . . 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 258 . . . . . 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 3151 . . . . 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 23075 . . . . . . 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 1120 . . . . . 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 2121 . . . . . . . 8 (𝑦 = 𝑀 β†’ (π‘Ž ∈ 𝑦 ↔ π‘Ž ∈ 𝑀))
22 elequ2 2121 . . . . . . . 8 (𝑦 = 𝑀 β†’ (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑀))
2321, 22bibi12d 345 . . . . . . 7 (𝑦 = 𝑀 β†’ ((π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀)))
2423cbvralvw 3225 . . . . . 6 (βˆ€π‘¦ ∈ 𝐽 (π‘Ž ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ βˆ€π‘€ ∈ 𝐽 (π‘Ž ∈ 𝑀 ↔ 𝑏 ∈ 𝑀))
2520, 24bitrdi 286 . . . . 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 258 . . . 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 3197 . . 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 23076 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
29 eleq1 2825 . . . . . . . . . 10 (𝑒 = (πΉβ€˜π‘Ž) β†’ (𝑒 ∈ 𝑧 ↔ (πΉβ€˜π‘Ž) ∈ 𝑧))
3029bibi1d 343 . . . . . . . . 9 (𝑒 = (πΉβ€˜π‘Ž) β†’ ((𝑒 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
3130ralbidv 3174 . . . . . . . 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 2740 . . . . . . . 8 (𝑒 = (πΉβ€˜π‘Ž) β†’ (𝑒 = 𝑣 ↔ (πΉβ€˜π‘Ž) = 𝑣))
3331, 32imbi12d 344 . . . . . . 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 3174 . . . . . 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 7038 . . . . 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 2825 . . . . . . . . . 10 (𝑣 = (πΉβ€˜π‘) β†’ (𝑣 ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧))
3736bibi2d 342 . . . . . . . . 9 (𝑣 = (πΉβ€˜π‘) β†’ (((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((πΉβ€˜π‘Ž) ∈ 𝑧 ↔ (πΉβ€˜π‘) ∈ 𝑧)))
3837ralbidv 3174 . . . . . . . 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 2748 . . . . . . . 8 (𝑣 = (πΉβ€˜π‘) β†’ ((πΉβ€˜π‘Ž) = 𝑣 ↔ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘)))
4038, 39imbi12d 344 . . . . . . 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 7038 . . . . . 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 3174 . . . . 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 278 . . . 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 256 . 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 23078 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
47 ist0-2 22695 . . 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 256 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3064  {crab 3407   ↦ cmpt 5188  ran crn 5634   β€œ cima 5636   Fn wfn 6491  β€˜cfv 6496  TopOnctopon 22259  Kol2ct0 22657  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-qtop 17389  df-top 22243  df-topon 22260  df-t0 22664  df-kq 23045
This theorem is referenced by:  kqt0  23097  t0kq  23169
  Copyright terms: Public domain W3C validator