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Theorem kqffn 23099
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 4041 . . . . 5 {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦} βŠ† 𝐽
2 elpw2g 5305 . . . . 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 7066 . 2 (𝐽 ∈ 𝑉 β†’ 𝐹:π‘‹βŸΆπ’« 𝐽)
76ffnd 6673 1 (𝐽 ∈ 𝑉 β†’ 𝐹 Fn 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406   βŠ† wss 3914  π’« cpw 4564   ↦ cmpt 5192   Fn wfn 6495
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-fun 6502  df-fn 6503  df-f 6504
This theorem is referenced by:  kqtopon  23101  kqid  23102  ist0-4  23103  kqfvima  23104  kqsat  23105  kqdisj  23106  kqcldsat  23107  kqopn  23108  kqcld  23109  kqt0lem  23110  isr0  23111  r0cld  23112  regr1lem2  23114  kqreglem1  23115  kqreglem2  23116  kqnrmlem1  23117  kqnrmlem2  23118
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