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Theorem kqffn 23734
Description: The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqffn (𝐽𝑉𝐹 Fn 𝑋)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦   𝑥,𝑉
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem kqffn
StepHypRef Expression
1 ssrab2 4079 . . . . 5 {𝑦𝐽𝑥𝑦} ⊆ 𝐽
2 elpw2g 5332 . . . . 5 (𝐽𝑉 → ({𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽 ↔ {𝑦𝐽𝑥𝑦} ⊆ 𝐽))
31, 2mpbiri 258 . . . 4 (𝐽𝑉 → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
43adantr 480 . . 3 ((𝐽𝑉𝑥𝑋) → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fmptd 7133 . 2 (𝐽𝑉𝐹:𝑋⟶𝒫 𝐽)
76ffnd 6736 1 (𝐽𝑉𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435  wss 3950  𝒫 cpw 4599  cmpt 5224   Fn wfn 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  kqtopon  23736  kqid  23737  ist0-4  23738  kqfvima  23739  kqsat  23740  kqdisj  23741  kqcldsat  23742  kqopn  23743  kqcld  23744  kqt0lem  23745  isr0  23746  r0cld  23747  regr1lem2  23749  kqreglem1  23750  kqreglem2  23751  kqnrmlem1  23752  kqnrmlem2  23753
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