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Theorem kqfeq 22260
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 22259 . . . 4 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
323adant3 1124 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
41kqfval 22259 . . . 4 ((𝐽𝑉𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
543adant2 1123 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
63, 5eqeq12d 2834 . 2 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦}))
7 rabbi 3381 . 2 (∀𝑦𝐽 (𝐴𝑦𝐵𝑦) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦})
86, 7syl6bbr 290 1 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  cmpt 5137  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  ist0-4  22265  kqfvima  22266  kqt0lem  22272  isr0  22273  r0cld  22274  regr1lem2  22276
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