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Theorem kqfeq 22327
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 22326 . . . 4 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
323adant3 1129 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
41kqfval 22326 . . . 4 ((𝐽𝑉𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
543adant2 1128 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
63, 5eqeq12d 2838 . 2 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦}))
7 rabbi 3364 . 2 (∀𝑦𝐽 (𝐴𝑦𝐵𝑦) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦})
86, 7syl6bbr 292 1 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114  wral 3130  {crab 3134  cmpt 5122  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342
This theorem is referenced by:  ist0-4  22332  kqfvima  22333  kqt0lem  22339  isr0  22340  r0cld  22341  regr1lem2  22343
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