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Theorem kqffn 22328
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 4042 . . . . 5 {𝑦𝐽𝑥𝑦} ⊆ 𝐽
2 elpw2g 5234 . . . . 5 (𝐽𝑉 → ({𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽 ↔ {𝑦𝐽𝑥𝑦} ⊆ 𝐽))
31, 2mpbiri 261 . . . 4 (𝐽𝑉 → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
43adantr 484 . . 3 ((𝐽𝑉𝑥𝑋) → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fmptd 6867 . 2 (𝐽𝑉𝐹:𝑋⟶𝒫 𝐽)
76ffnd 6504 1 (𝐽𝑉𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  wss 3919  𝒫 cpw 4522  cmpt 5133   Fn wfn 6339
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352
This theorem is referenced by:  kqtopon  22330  kqid  22331  ist0-4  22332  kqfvima  22333  kqsat  22334  kqdisj  22335  kqcldsat  22336  kqopn  22337  kqcld  22338  kqt0lem  22339  isr0  22340  r0cld  22341  regr1lem2  22343  kqreglem1  22344  kqreglem2  22345  kqnrmlem1  22346  kqnrmlem2  22347
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