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Theorem kqfval 23090
Description: Value of the function appearing in df-kq 23061. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqfval ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝑉
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem kqfval
StepHypRef Expression
1 id 22 . 2 (𝐴𝑋𝐴𝑋)
2 rabexg 5293 . 2 (𝐽𝑉 → {𝑦𝐽𝐴𝑦} ∈ V)
3 eleq1 2826 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
43rabbidv 3418 . . 3 (𝑥 = 𝐴 → {𝑦𝐽𝑥𝑦} = {𝑦𝐽𝐴𝑦})
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fvmptg 6951 . 2 ((𝐴𝑋 ∧ {𝑦𝐽𝐴𝑦} ∈ V) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
71, 2, 6syl2anr 598 1 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  cmpt 5193  cfv 6501
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  kqfeq  23091  isr0  23104
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