Theorem nfmpt2 5676

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

Hypotheses

Ref	Expression
nfmpt2.1	⊢ ℲzA
nfmpt2.2	⊢ ℲzB
nfmpt2.3	⊢ ℲzC

Assertion

Ref	Expression
nfmpt2	⊢ Ⅎz(x ∈ A, y ∈ 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 ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = C)}
2		nfmpt2.1	. . . . . 6 ⊢ ℲzA
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎz x ∈ A
4		nfmpt2.2	. . . . . 6 ⊢ ℲzB
5	4	nfcri 2484	. . . . 5 ⊢ Ⅎz y ∈ B
6	3, 5	nfan 1824	. . . 4 ⊢ Ⅎz(x ∈ A ∧ y ∈ B)
7		nfmpt2.3	. . . . 5 ⊢ ℲzC
8	7	nfeq2 2501	. . . 4 ⊢ Ⅎz w = C
9	6, 8	nfan 1824	. . 3 ⊢ Ⅎz((x ∈ A ∧ y ∈ B) ∧ w = C)
10	9	nfoprab 5550	. 2 ⊢ Ⅎz{⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = C)}
11	1, 10	nfcxfr 2487	1 ⊢ Ⅎz(x ∈ A, y ∈ B ↦ C)

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 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)