Theorem nfmpo 6073

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

Hypotheses

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

Assertion

Ref	Expression
nfmpo	|- 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 nfmpo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 6006	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
2		nfmpo.1	. . . . . 6 |- F/_ z A
3	2	nfcri 2366	. . . . 5 |- F/ z x e. A
4		nfmpo.2	. . . . . 6 |- F/_ z B
5	4	nfcri 2366	. . . . 5 |- F/ z y e. B
6	3, 5	nfan 1611	. . . 4 |- F/ z ( x e. A /\ y e. B )
7		nfmpo.3	. . . . 5 |- F/_ z C
8	7	nfeq2 2384	. . . 4 |- F/ z w = C
9	6, 8	nfan 1611	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ w = C )
10	9	nfoprab 6056	. 2 |- F/_ z { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
11	1, 10	nfcxfr 2369	1 |- F/_ z ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   F/_wnfc 2359   {coprab 6002   e. cmpo 6003

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-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-oprab 6005  df-mpo 6006

This theorem is referenced by:  nfof  6224  nfseq  10679