Theorem nfmpt2 5676

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

Hypotheses

Ref	Expression
nfmpt2.1	|- F/_zA
nfmpt2.2	|- F/_zB
nfmpt2.3	|- F/_zC

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 5655	. 2 |- (x e. A, y e. B |-> C) = {<.<.x, y>., w>. | ((x e. A /\ y e. B) /\ w = C)}
2		nfmpt2.1	. . . . . 6 |- F/_zA
3	2	nfcri 2484	. . . . 5 |- F/z x e. A
4		nfmpt2.2	. . . . . 6 |- F/_zB
5	4	nfcri 2484	. . . . 5 |- F/z y e. B
6	3, 5	nfan 1824	. . . 4 |- F/z(x e. A /\ y e. B)
7		nfmpt2.3	. . . . 5 |- F/_zC
8	7	nfeq2 2501	. . . 4 |- F/z w = C
9	6, 8	nfan 1824	. . 3 |- F/z((x e. A /\ y e. B) /\ w = C)
10	9	nfoprab 5550	. 2 |- F/_z{<.<.x, y>., w>. | ((x e. A /\ y e. B) /\ w = C)}
11	1, 10	nfcxfr 2487	1 |- F/_z(x e. A, y e. B |-> C)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710   F/_wnfc 2477  {coprab 5528   |-> cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-oprab 5529  df-mpt2 5655

This theorem is referenced by: (None)