Theorem riota5 5982

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 2373	. 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 5981	1 |- ( ph -> ( iota_ x e. A ps ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   iota_crio 5953

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  f1ocnvfv3  5990  caucvgrelemrec  11490  sqrt0  11515  sqrtsq  11555  dfgcd3  12531