Theorem nfmpt 4081

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 4052	. 2 |- ( y e. A |-> B ) = { <. y , z >. | ( y e. A /\ z = B ) }
2		nfmpt.1	. . . . 5 |- F/_ x A
3	2	nfcri 2306	. . . 4 |- F/ x y e. A
4		nfmpt.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2324	. . . 4 |- F/ x z = B
6	3, 5	nfan 1558	. . 3 |- F/ x ( y e. A /\ z = B )
7	6	nfopab 4057	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z = B ) }
8	1, 7	nfcxfr 2309	1 |- F/_ x ( y e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   F/_wnfc 2299   {copab 4049   |-> cmpt 4050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-opab 4051  df-mpt 4052

This theorem is referenced by:  nfof  6066  nffrec  6375  mapxpen  6826  nfsum1  11319  nfsum  11320  nfcprod1  11517  nfcprod  11518  ctiunct  12395  fsumcncntop  13350  limcmpted  13426