Theorem riota5 5858

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

Step	Hyp	Ref	Expression
1		nfcvd 2320	. 2 |- ( ph -> F/_ x B )
2		riota5.1	. 2 |- ( ph -> B e. A )
3		riota5.2	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) )
4	1, 2, 3	riota5f 5857	1 |- ( ph -> ( iota_ x e. A ps ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   iota_crio 5832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812  df-iota 5180  df-riota 5833

This theorem is referenced by:  f1ocnvfv3  5866  caucvgrelemrec  10990  sqrt0  11015  sqrtsq  11055  dfgcd3  12013