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Theorem kqt0lem 21780
Description: Lemma for kqt0 21790. (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 21778 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
32adantlr 695 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
4 eleq2 2842 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
5 eleq2 2842 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → ((𝐹𝑏) ∈ 𝑧 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
64, 5bibi12d 335 . . . . . . . . 9 (𝑧 = (𝐹𝑤) → (((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
76rspcv 3461 . . . . . . . 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 21774 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1093expa 1138 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑎𝑋) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
1110adantrr 697 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝑤 ↔ (𝐹𝑎) ∈ (𝐹𝑤)))
121kqfvima 21774 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
13123expa 1138 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ 𝑏𝑋) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1413adantrl 696 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → (𝑏𝑤 ↔ (𝐹𝑏) ∈ (𝐹𝑤)))
1511, 14bibi12d 335 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
1615an32s 632 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → ((𝑎𝑤𝑏𝑤) ↔ ((𝐹𝑎) ∈ (𝐹𝑤) ↔ (𝐹𝑏) ∈ (𝐹𝑤))))
178, 16sylibrd 250 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑤𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝑎𝑤𝑏𝑤)))
1817ralrimdva 3121 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
191kqfeq 21768 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
20193expb 1140 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑦𝐽 (𝑎𝑦𝑏𝑦)))
21 elequ2 2162 . . . . . . . 8 (𝑦 = 𝑤 → (𝑎𝑦𝑎𝑤))
22 elequ2 2162 . . . . . . . 8 (𝑦 = 𝑤 → (𝑏𝑦𝑏𝑤))
2321, 22bibi12d 335 . . . . . . 7 (𝑦 = 𝑤 → ((𝑎𝑦𝑏𝑦) ↔ (𝑎𝑤𝑏𝑤)))
2423cbvralv 3324 . . . . . 6 (∀𝑦𝐽 (𝑎𝑦𝑏𝑦) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤))
2520, 24syl6bb 277 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑎) = (𝐹𝑏) ↔ ∀𝑤𝐽 (𝑎𝑤𝑏𝑤)))
2618, 25sylibrd 250 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
2726ralrimivva 3123 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎𝑋𝑏𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧) → (𝐹𝑎) = (𝐹𝑏)))
281kqffn 21769 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29 eleq1 2841 . . . . . . . . . 10 (𝑢 = (𝐹𝑎) → (𝑢𝑧 ↔ (𝐹𝑎) ∈ 𝑧))
3029bibi1d 333 . . . . . . . . 9 (𝑢 = (𝐹𝑎) → ((𝑢𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
3130ralbidv 3138 . . . . . . . 8 (𝑢 = (𝐹𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧)))
32 eqeq1 2778 . . . . . . . 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 3138 . . . . . 6 (𝑢 = (𝐹𝑎) → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) → (𝐹𝑎) = 𝑣)))
3534ralrn 6522 . . . . 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 2841 . . . . . . . . . 10 (𝑣 = (𝐹𝑏) → (𝑣𝑧 ↔ (𝐹𝑏) ∈ 𝑧))
3736bibi2d 332 . . . . . . . . 9 (𝑣 = (𝐹𝑏) → (((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
3837ralbidv 3138 . . . . . . . 8 (𝑣 = (𝐹𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧𝑣𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹𝑎) ∈ 𝑧 ↔ (𝐹𝑏) ∈ 𝑧)))
39 eqeq2 2785 . . . . . . . 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 6522 . . . . . 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 3138 . . . . 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 269 . . . 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 248 . 2 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣))
461kqtopon 21771 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47 ist0-2 21389 . . 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 248 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wral 3064  {crab 3068  cmpt 4876  ran crn 5264  cima 5266   Fn wfn 6037  cfv 6042  TopOnctopon 20955  Kol2ct0 21351  KQckq 21737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-qtop 16395  df-top 20939  df-topon 20956  df-t0 21358  df-kq 21738
This theorem is referenced by:  kqt0  21790  t0kq  21862
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