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Theorem kqt0lem 23760
Description: Lemma for kqt0 23770. (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 23758 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
32adantlr 715 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
4 eleq2 2828 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
5 eleq2 2828 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑏) ∈ 𝑧 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
64, 5bibi12d 345 . . . . . . . . 9 (𝑧 = (𝐹𝑤) → (((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
76rspcv 3618 . . . . . . . 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 23754 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1093expa 1117 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1110adantrr 717 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
121kqfvima 23754 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
13123expa 1117 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1413adantrl 716 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1511, 14bibi12d 345 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
1615an32s 652 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
178, 16sylibrd 259 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝑎𝑤𝑏𝑤)))
1817ralrimdva 3152 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
191kqfeq 23748 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
20193expb 1119 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
21 elequ2 2121 . . . . . . . 8 (𝑦 = 𝑤 → (𝑎𝑦𝑎𝑤))
22 elequ2 2121 . . . . . . . 8 (𝑦 = 𝑤 → (𝑏𝑦𝑏𝑤))
2321, 22bibi12d 345 . . . . . . 7 (𝑦 = 𝑤 → ((𝑎𝑦𝑏𝑦) ↔ (𝑎𝑤𝑏𝑤)))
2423cbvralvw 3235 . . . . . 6 (∀𝑦𝐽 (𝑎𝑦𝑏𝑦) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤))
2520, 24bitrdi 287 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
2618, 25sylibrd 259 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
2726ralrimivva 3200 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
281kqffn 23749 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29 eleq1 2827 . . . . . . . . . 10 (𝑢 = (𝐹𝑎) → (𝑢𝑧 ↔ (𝐹𝑎) ∈ 𝑧))
3029bibi1d 343 . . . . . . . . 9 (𝑢 = (𝐹𝑎) → ((𝑢𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
3130ralbidv 3176 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
32 eqeq1 2739 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (𝑢 = 𝑣 ↔ (𝐹𝑎) = 𝑣))
3331, 32imbi12d 344 . . . . . . 7 (𝑢 = (𝐹𝑎) → ((∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3433ralbidv 3176 . . . . . 6 (𝑢 = (𝐹𝑎) → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3534ralrn 7108 . . . . 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 2827 . . . . . . . . . 10 (𝑣 = (𝐹𝑏) → (𝑣𝑧 ↔ (𝐹𝑏) ∈ 𝑧))
3736bibi2d 342 . . . . . . . . 9 (𝑣 = (𝐹𝑏) → (((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
3837ralbidv 3176 . . . . . . . 8 (𝑣 = (𝐹𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
39 eqeq2 2747 . . . . . . . 8 (𝑣 = (𝐹𝑏) → ((𝐹𝑎) = 𝑣 ↔ (𝐹𝑎) = (𝐹𝑏)))
4038, 39imbi12d 344 . . . . . . 7 (𝑣 = (𝐹𝑏) → ((∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏))))
4140ralrn 7108 . . . . . 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 3176 . . . . 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 279 . . . 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 257 . 2 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣))
461kqtopon 23751 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47 ist0-2 23368 . . 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 257 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cmpt 5231  ran crn 5690  cima 5692   Fn wfn 6558  cfv 6563  TopOnctopon 22932  Kol2ct0 23330  KQckq 23717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17554  df-top 22916  df-topon 22933  df-t0 23337  df-kq 23718
This theorem is referenced by:  kqt0  23770  t0kq  23842
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