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Theorem kqt0lem 21449
Description: Lemma for kqt0 21459. (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 21447 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
32adantlr 750 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
4 eleq2 2687 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
5 eleq2 2687 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑏) ∈ 𝑧 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
64, 5bibi12d 335 . . . . . . . . 9 (𝑧 = (𝐹𝑤) → (((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
76rspcv 3291 . . . . . . . 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 21443 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1093expa 1262 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1110adantrr 752 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
121kqfvima 21443 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
13123expa 1262 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1413adantrl 751 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1511, 14bibi12d 335 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
1615an32s 845 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
178, 16sylibrd 249 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝑎𝑤𝑏𝑤)))
1817ralrimdva 2963 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
191kqfeq 21437 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
20193expb 1263 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
21 elequ2 2001 . . . . . . . 8 (𝑦 = 𝑤 → (𝑎𝑦𝑎𝑤))
22 elequ2 2001 . . . . . . . 8 (𝑦 = 𝑤 → (𝑏𝑦𝑏𝑤))
2321, 22bibi12d 335 . . . . . . 7 (𝑦 = 𝑤 → ((𝑎𝑦𝑏𝑦) ↔ (𝑎𝑤𝑏𝑤)))
2423cbvralv 3159 . . . . . 6 (∀𝑦𝐽 (𝑎𝑦𝑏𝑦) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤))
2520, 24syl6bb 276 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
2618, 25sylibrd 249 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
2726ralrimivva 2965 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
281kqffn 21438 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29 eleq1 2686 . . . . . . . . . 10 (𝑢 = (𝐹𝑎) → (𝑢𝑧 ↔ (𝐹𝑎) ∈ 𝑧))
3029bibi1d 333 . . . . . . . . 9 (𝑢 = (𝐹𝑎) → ((𝑢𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
3130ralbidv 2980 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
32 eqeq1 2625 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (𝑢 = 𝑣 ↔ (𝐹𝑎) = 𝑣))
3331, 32imbi12d 334 . . . . . . 7 (𝑢 = (𝐹𝑎) → ((∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3433ralbidv 2980 . . . . . 6 (𝑢 = (𝐹𝑎) → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3534ralrn 6318 . . . . 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 2686 . . . . . . . . . 10 (𝑣 = (𝐹𝑏) → (𝑣𝑧 ↔ (𝐹𝑏) ∈ 𝑧))
3736bibi2d 332 . . . . . . . . 9 (𝑣 = (𝐹𝑏) → (((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
3837ralbidv 2980 . . . . . . . 8 (𝑣 = (𝐹𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
39 eqeq2 2632 . . . . . . . 8 (𝑣 = (𝐹𝑏) → ((𝐹𝑎) = 𝑣 ↔ (𝐹𝑎) = (𝐹𝑏)))
4038, 39imbi12d 334 . . . . . . 7 (𝑣 = (𝐹𝑏) → ((∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4140ralrn 6318 . . . . . 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 2980 . . . . 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 268 . . . 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 247 . 2 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣))
461kqtopon 21440 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47 ist0-2 21058 . . 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 247 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cmpt 4673  ran crn 5075  cima 5077   Fn wfn 5842  cfv 5847  TopOnctopon 20618  Kol2ct0 21020  KQckq 21406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-qtop 16088  df-top 20621  df-topon 20623  df-t0 21027  df-kq 21407
This theorem is referenced by:  kqt0  21459  t0kq  21531
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