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Theorem kqffn 23749
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 4090 . . . . 5 {𝑦𝐽𝑥𝑦} ⊆ 𝐽
2 elpw2g 5339 . . . . 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 7134 . 2 (𝐽𝑉𝐹:𝑋⟶𝒫 𝐽)
76ffnd 6738 1 (𝐽𝑉𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  wss 3963  𝒫 cpw 4605  cmpt 5231   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  kqtopon  23751  kqid  23752  ist0-4  23753  kqfvima  23754  kqsat  23755  kqdisj  23756  kqcldsat  23757  kqopn  23758  kqcld  23759  kqt0lem  23760  isr0  23761  r0cld  23762  regr1lem2  23764  kqreglem1  23765  kqreglem2  23766  kqnrmlem1  23767  kqnrmlem2  23768
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