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Theorem kqffn 23228
Description: The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqffn (𝐽 ∈ 𝑉 β†’ 𝐹 Fn 𝑋)
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦   π‘₯,𝑉
Allowed substitution hints:   𝐹(π‘₯,𝑦)   𝑉(𝑦)

Proof of Theorem kqffn
StepHypRef Expression
1 ssrab2 4077 . . . . 5 {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} βŠ† 𝐽
2 elpw2g 5344 . . . . 5 (𝐽 ∈ 𝑉 β†’ ({𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} βŠ† 𝐽))
31, 2mpbiri 257 . . . 4 (𝐽 ∈ 𝑉 β†’ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} ∈ 𝒫 𝐽)
43adantr 481 . . 3 ((𝐽 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} ∈ 𝒫 𝐽)
5 kqval.2 . . 3 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
64, 5fmptd 7113 . 2 (𝐽 ∈ 𝑉 β†’ 𝐹:π‘‹βŸΆπ’« 𝐽)
76ffnd 6718 1 (𝐽 ∈ 𝑉 β†’ 𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3948  π’« cpw 4602   ↦ cmpt 5231   Fn wfn 6538
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  kqtopon  23230  kqid  23231  ist0-4  23232  kqfvima  23233  kqsat  23234  kqdisj  23235  kqcldsat  23236  kqopn  23237  kqcld  23238  kqt0lem  23239  isr0  23240  r0cld  23241  regr1lem2  23243  kqreglem1  23244  kqreglem2  23245  kqnrmlem1  23246  kqnrmlem2  23247
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