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Theorem kqt0lem 23731
Description: Lemma for kqt0 23741. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqt0lem (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqt0lem
Dummy variables 𝑤 𝑧 𝑎 𝑏 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . . . . 10 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqopn 23729 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
32adantlr 713 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
4 eleq2 2815 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
5 eleq2 2815 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑏) ∈ 𝑧 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
64, 5bibi12d 344 . . . . . . . . 9 (𝑧 = (𝐹𝑤) → (((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
76rspcv 3604 . . . . . . . 8 ((𝐹𝑤) ∈ (KQ‘𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
83, 7syl 17 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
91kqfvima 23725 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1093expa 1115 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1110adantrr 715 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
121kqfvima 23725 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
13123expa 1115 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1413adantrl 714 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1511, 14bibi12d 344 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
1615an32s 650 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
178, 16sylibrd 258 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝑎𝑤𝑏𝑤)))
1817ralrimdva 3144 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
191kqfeq 23719 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
20193expb 1117 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
21 elequ2 2114 . . . . . . . 8 (𝑦 = 𝑤 → (𝑎𝑦𝑎𝑤))
22 elequ2 2114 . . . . . . . 8 (𝑦 = 𝑤 → (𝑏𝑦𝑏𝑤))
2321, 22bibi12d 344 . . . . . . 7 (𝑦 = 𝑤 → ((𝑎𝑦𝑏𝑦) ↔ (𝑎𝑤𝑏𝑤)))
2423cbvralvw 3225 . . . . . 6 (∀𝑦𝐽 (𝑎𝑦𝑏𝑦) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤))
2520, 24bitrdi 286 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
2618, 25sylibrd 258 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
2726ralrimivva 3191 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
281kqffn 23720 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29 eleq1 2814 . . . . . . . . . 10 (𝑢 = (𝐹𝑎) → (𝑢𝑧 ↔ (𝐹𝑎) ∈ 𝑧))
3029bibi1d 342 . . . . . . . . 9 (𝑢 = (𝐹𝑎) → ((𝑢𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
3130ralbidv 3168 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
32 eqeq1 2730 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (𝑢 = 𝑣 ↔ (𝐹𝑎) = 𝑣))
3331, 32imbi12d 343 . . . . . . 7 (𝑢 = (𝐹𝑎) → ((∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3433ralbidv 3168 . . . . . 6 (𝑢 = (𝐹𝑎) → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3534ralrn 7102 . . . . 5 (𝐹 Fn 𝑋 → (∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎𝑋𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
36 eleq1 2814 . . . . . . . . . 10 (𝑣 = (𝐹𝑏) → (𝑣𝑧 ↔ (𝐹𝑏) ∈ 𝑧))
3736bibi2d 341 . . . . . . . . 9 (𝑣 = (𝐹𝑏) → (((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
3837ralbidv 3168 . . . . . . . 8 (𝑣 = (𝐹𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
39 eqeq2 2738 . . . . . . . 8 (𝑣 = (𝐹𝑏) → ((𝐹𝑎) = 𝑣 ↔ (𝐹𝑎) = (𝐹𝑏)))
4038, 39imbi12d 343 . . . . . . 7 (𝑣 = (𝐹𝑏) → ((∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4140ralrn 7102 . . . . . 6 (𝐹 Fn 𝑋 → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣) ↔ ∀𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4241ralbidv 3168 . . . . 5 (𝐹 Fn 𝑋 → (∀𝑎𝑋𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣) ↔ ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4335, 42bitrd 278 . . . 4 (𝐹 Fn 𝑋 → (∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4428, 43syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4527, 44mpbird 256 . 2 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣))
461kqtopon 23722 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47 ist0-2 23339 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Kol2 ↔ ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣)))
4846, 47syl 17 . 2 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Kol2 ↔ ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣)))
4945, 48mpbird 256 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  cmpt 5236  ran crn 5683  cima 5685   Fn wfn 6549  cfv 6554  TopOnctopon 22903  Kol2ct0 23301  KQckq 23688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-qtop 17522  df-top 22887  df-topon 22904  df-t0 23308  df-kq 23689
This theorem is referenced by:  kqt0  23741  t0kq  23813
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