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Theorem riota5 6292
Description: A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
riota5.1  |-  ( ph  ->  B  e.  A )
riota5.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  x  =  B ) )
Assertion
Ref Expression
riota5  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  B
)
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem riota5
StepHypRef Expression
1 nfcvd 2581 . 2  |-  ( ph  -> 
F/_ x B )
2 riota5.1 . 2  |-  ( ph  ->  B  e.  A )
3 riota5.2 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  x  =  B ) )
41, 2, 3riota5f 6291 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   iota_crio 6266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-reu 2778  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565  df-riota 6267
This theorem is referenced by:  f1ocnvfv3  6301  sqrt0  13305  lubid  16235  lubun  16368  odval2  17199  adjvalval  27588  xdivpnfrp  28409  xrsinvgval  28446  poimirlem6  31910  poimirlem7  31911  lub0N  32724  glb0N  32728  trlval2  33698  cdlemefrs32fva  33936  cdleme32fva  33973  cdlemg1a  34106  unxpwdom3  35923
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