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Theorem kqffn 21906
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 3914 . . . . 5 {𝑦𝐽𝑥𝑦} ⊆ 𝐽
2 elpw2g 5051 . . . . 5 (𝐽𝑉 → ({𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽 ↔ {𝑦𝐽𝑥𝑦} ⊆ 𝐽))
31, 2mpbiri 250 . . . 4 (𝐽𝑉 → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
43adantr 474 . . 3 ((𝐽𝑉𝑥𝑋) → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fmptd 6638 . 2 (𝐽𝑉𝐹:𝑋⟶𝒫 𝐽)
76ffnd 6283 1 (𝐽𝑉𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {crab 3121  wss 3798  𝒫 cpw 4380  cmpt 4954   Fn wfn 6122
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-ne 3000  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-pw 4382  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-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135
This theorem is referenced by:  kqtopon  21908  kqid  21909  ist0-4  21910  kqfvima  21911  kqsat  21912  kqdisj  21913  kqcldsat  21914  kqopn  21915  kqcld  21916  kqt0lem  21917  isr0  21918  r0cld  21919  regr1lem2  21921  kqreglem1  21922  kqreglem2  21923  kqnrmlem1  21924  kqnrmlem2  21925
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