Theorem nfmpt2 6724

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 6655	. 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 2758	. . . . 5 |- F/ z x e. A
4		nfmpt2.2	. . . . . 6 |- F/_ z B
5	4	nfcri 2758	. . . . 5 |- F/ z y e. B
6	3, 5	nfan 1828	. . . 4 |- F/ z ( x e. A /\ y e. B )
7		nfmpt2.3	. . . . 5 |- F/_ z C
8	7	nfeq2 2780	. . . 4 |- F/ z w = C
9	6, 8	nfan 1828	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ w = C )
10	9	nfoprab 6707	. 2 |- F/_ z { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = C ) }
11	1, 10	nfcxfr 2762	1 |- F/_ z ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  el2mpt2csbcl  7250  nfseq  12811  ptbasfi  21384  numclwlk1lem2  27230  sdclem1  33539  fmuldfeqlem1  39814  stoweidlem51  40268  vonicc  40899