Theorem nfmpo 6016

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 5951	. 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 2342	. . . . 5 |- F/ z x e. A
4		nfmpo.2	. . . . . 6 |- F/_ z B
5	4	nfcri 2342	. . . . 5 |- F/ z y e. B
6	3, 5	nfan 1588	. . . 4 |- F/ z ( x e. A /\ y e. B )
7		nfmpo.3	. . . . 5 |- F/_ z C
8	7	nfeq2 2360	. . . 4 |- F/ z w = C
9	6, 8	nfan 1588	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ w = C )
10	9	nfoprab 5999	. 2 |- F/_ z { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
11	1, 10	nfcxfr 2345	1 |- F/_ z ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   F/_wnfc 2335   {coprab 5947   e. cmpo 5948

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-oprab 5950  df-mpo 5951

This theorem is referenced by:  nfof  6166  nfseq  10604