Theorem nfmpt2 5593

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Hypotheses

Ref	Expression
nfmpt2.1	|- F/_ z A
nfmpt2.2	|- F/_ z B
nfmpt2.3	|- F/_ z C

Assertion

Ref	Expression
nfmpt2	|- F/_ z ( x e. A , y e. B |-> C )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   A( x, y, z)   B( x, y, z)   C( x, y, z)

Proof of Theorem nfmpt2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 5537	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
2		nfmpt2.1	. . . . . 6 |- F/_ z A
3	2	nfcri 2213	. . . . 5 |- F/ z x e. A
4		nfmpt2.2	. . . . . 6 |- F/_ z B
5	4	nfcri 2213	. . . . 5 |- F/ z y e. B
6	3, 5	nfan 1497	. . . 4 |- F/ z ( x e. A /\ y e. B )
7		nfmpt2.3	. . . . 5 |- F/_ z C
8	7	nfeq2 2230	. . . 4 |- F/ z w = C
9	6, 8	nfan 1497	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ w = C )
10	9	nfoprab 5577	. 2 |- F/_ z { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
11	1, 10	nfcxfr 2216	1 |- F/_ z ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   F/_wnfc 2206   {coprab 5533   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  nfiseq  9438