Theorem riota5 5899

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 2337	. 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 5898	1 |- ( ph -> ( iota_ x e. A ps ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   iota_crio 5872

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215  df-riota 5873

This theorem is referenced by:  f1ocnvfv3  5907  caucvgrelemrec  11123  sqrt0  11148  sqrtsq  11188  dfgcd3  12147