Theorem nfmpt 4186

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 4157	. 2 |- ( y e. A |-> B ) = { <. y , z >. | ( y e. A /\ z = B ) }
2		nfmpt.1	. . . . 5 |- F/_ x A
3	2	nfcri 2369	. . . 4 |- F/ x y e. A
4		nfmpt.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2387	. . . 4 |- F/ x z = B
6	3, 5	nfan 1614	. . 3 |- F/ x ( y e. A /\ z = B )
7	6	nfopab 4162	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z = B ) }
8	1, 7	nfcxfr 2372	1 |- F/_ x ( y e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   F/_wnfc 2362   {copab 4154   |-> cmpt 4155

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-opab 4156  df-mpt 4157

This theorem is referenced by:  nfof  6250  nffrec  6605  mapxpen  7077  nfsum1  11996  nfsum  11997  nfcprod1  12195  nfcprod  12196  ctiunct  13141  fsumcncntop  15378  limcmpted  15474  dvmptfsum  15536