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Theorem kqfeq 23732
Description: Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqfeq ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝑉
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem kqfeq
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqfval 23731 . . . 4 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
323adant3 1133 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
41kqfval 23731 . . . 4 ((𝐽𝑉𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
543adant2 1132 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
63, 5eqeq12d 2753 . 2 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦}))
7 rabbi 3467 . 2 (∀𝑦𝐽 (𝐴𝑦𝐵𝑦) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦})
86, 7bitr4di 289 1 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  cmpt 5225  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  ist0-4  23737  kqfvima  23738  kqt0lem  23744  isr0  23745  r0cld  23746  regr1lem2  23748
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