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Theorem kqffn 23229
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 4078 . . . . 5 {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} βŠ† 𝐽
2 elpw2g 5345 . . . . 5 (𝐽 ∈ 𝑉 β†’ ({𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} βŠ† 𝐽))
31, 2mpbiri 258 . . . 4 (𝐽 ∈ 𝑉 β†’ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} ∈ 𝒫 𝐽)
43adantr 482 . . 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 7114 . 2 (𝐽 ∈ 𝑉 β†’ 𝐹:π‘‹βŸΆπ’« 𝐽)
76ffnd 6719 1 (𝐽 ∈ 𝑉 β†’ 𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949  π’« cpw 4603   ↦ cmpt 5232   Fn wfn 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  kqtopon  23231  kqid  23232  ist0-4  23233  kqfvima  23234  kqsat  23235  kqdisj  23236  kqcldsat  23237  kqopn  23238  kqcld  23239  kqt0lem  23240  isr0  23241  r0cld  23242  regr1lem2  23244  kqreglem1  23245  kqreglem2  23246  kqnrmlem1  23247  kqnrmlem2  23248
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