Theorem riota5 5834

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 2313	. 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 5833	1 |- ( ph -> ( iota_ x e. A ps ) = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   iota_crio 5808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  f1ocnvfv3  5842  caucvgrelemrec  10943  sqrt0  10968  sqrtsq  11008  dfgcd3  11965