Theorem nfmpt 4181

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

Hypotheses

Ref	Expression
nfmpt.1	|- F/_ x A
nfmpt.2	|- F/_ x B

Assertion

Ref	Expression
nfmpt	|- F/_ x ( y e. A |-> B )

Distinct variable group:   x, y

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

Proof of Theorem nfmpt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4152	. 2 |- ( y e. A |-> B ) = { <. y , z >. | ( y e. A /\ z = B ) }
2		nfmpt.1	. . . . 5 |- F/_ x A
3	2	nfcri 2368	. . . 4 |- F/ x y e. A
4		nfmpt.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2386	. . . 4 |- F/ x z = B
6	3, 5	nfan 1613	. . 3 |- F/ x ( y e. A /\ z = B )
7	6	nfopab 4157	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z = B ) }
8	1, 7	nfcxfr 2371	1 |- F/_ x ( y e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   F/_wnfc 2361   {copab 4149   |-> cmpt 4150

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-opab 4151  df-mpt 4152

This theorem is referenced by:  nfof  6240  nffrec  6561  mapxpen  7033  nfsum1  11916  nfsum  11917  nfcprod1  12114  nfcprod  12115  ctiunct  13060  fsumcncntop  15290  limcmpted  15386  dvmptfsum  15448