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Theorem kqfval 21466
Description: Value of the function appearing in df-kq 21437. (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 4782 . 2 (𝐽𝑉 → {𝑦𝐽𝐴𝑦} ∈ V)
3 eleq1 2686 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
43rabbidv 3181 . . 3 (𝑥 = 𝐴 → {𝑦𝐽𝑥𝑦} = {𝑦𝐽𝐴𝑦})
5 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
64, 5fvmptg 6247 . 2 ((𝐴𝑋 ∧ {𝑦𝐽𝐴𝑦} ∈ V) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
71, 2, 6syl2anr 495 1 ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  cmpt 4683  cfv 5857
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-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-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-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  kqfeq  21467  isr0  21480
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