MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqcld Structured version   Visualization version   GIF version
Theorem kqcld 23239
Description: The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqcld ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(KQβ€˜π½)))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)
Proof of Theorem kqcld
StepHypRef Expression
1 imassrn 6071 . . . 4 (𝐹 β€œ π‘ˆ) βŠ† ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ π‘ˆ) βŠ† ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
43kqcldsat 23237 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
5 simpr 486 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ ∈ (Clsdβ€˜π½))
64, 5eqeltrd 2834 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ (Clsdβ€˜π½))
73kqffn 23229 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
8 dffn4 6812 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
97, 8sylib 217 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
10 qtopcld 23217 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ((𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ ((𝐹 β€œ π‘ˆ) βŠ† ran 𝐹 ∧ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ (Clsdβ€˜π½))))
119, 10mpdan 686 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ ((𝐹 β€œ π‘ˆ) βŠ† ran 𝐹 ∧ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ (Clsdβ€˜π½))))
1211adantr 482 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ ((𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ ((𝐹 β€œ π‘ˆ) βŠ† ran 𝐹 ∧ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ (Clsdβ€˜π½))))
132, 6, 12mpbir2and 712 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(𝐽 qTop 𝐹)))
143kqval 23230 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
1514adantr 482 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
1615fveq2d 6896 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (Clsdβ€˜(KQβ€˜π½)) = (Clsdβ€˜(𝐽 qTop 𝐹)))
1713, 16eleqtrrd 2837 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ π‘ˆ) ∈ (Clsdβ€˜(KQβ€˜π½)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949   ↦ cmpt 5232  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680   Fn wfn 6539  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   qTop cqtop 17449  TopOnctopon 22412  Clsdccld 22520  KQckq 23197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453  df-top 22396  df-topon 22413  df-cld 22523  df-kq 23198
This theorem is referenced by:  kqreglem1  23245  kqnrmlem1  23247  kqnrmlem2  23248
  Copyright terms: Public domain W3C validator