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Theorem kqffn 21468
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 3672 . . . . 5 {𝑦𝐽𝑥𝑦} ⊆ 𝐽
2 elpw2g 4797 . . . . 5 (𝐽𝑉 → ({𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽 ↔ {𝑦𝐽𝑥𝑦} ⊆ 𝐽))
31, 2mpbiri 248 . . . 4 (𝐽𝑉 → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
43adantr 481 . . 3 ((𝐽𝑉𝑥𝑋) → {𝑦𝐽𝑥𝑦} ∈ 𝒫 𝐽)
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fmptd 6351 . 2 (𝐽𝑉𝐹:𝑋⟶𝒫 𝐽)
7 ffn 6012 . 2 (𝐹:𝑋⟶𝒫 𝐽𝐹 Fn 𝑋)
86, 7syl 17 1 (𝐽𝑉𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2912  wss 3560  𝒫 cpw 4136  cmpt 4683   Fn wfn 5852  wf 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865
This theorem is referenced by:  kqtopon  21470  kqid  21471  ist0-4  21472  kqfvima  21473  kqsat  21474  kqdisj  21475  kqcldsat  21476  kqopn  21477  kqcld  21478  kqt0lem  21479  isr0  21480  r0cld  21481  regr1lem2  21483  kqreglem1  21484  kqreglem2  21485  kqnrmlem1  21486  kqnrmlem2  21487
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