MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqfeq Structured version   Visualization version   GIF version

Theorem kqfeq 21905
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 21904 . . . 4 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
323adant3 1166 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
41kqfval 21904 . . . 4 ((𝐽𝑉𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
543adant2 1165 . . 3 ((𝐽𝑉𝐴𝑋𝐵𝑋) → (𝐹𝐵) = {𝑦𝐽𝐵𝑦})
63, 5eqeq12d 2840 . 2 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦}))
7 rabbi 3331 . 2 (∀𝑦𝐽 (𝐴𝑦𝐵𝑦) ↔ {𝑦𝐽𝐴𝑦} = {𝑦𝐽𝐵𝑦})
86, 7syl6bbr 281 1 ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111   = wceq 1656  wcel 2164  wral 3117  {crab 3121  cmpt 4954  cfv 6127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135
This theorem is referenced by:  ist0-4  21910  kqfvima  21911  kqt0lem  21917  isr0  21918  r0cld  21919  regr1lem2  21921
  Copyright terms: Public domain W3C validator