Theorem nfmpt 5672

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

Hypotheses

Ref	Expression
nfmpt.1	⊢ ℲxA
nfmpt.2	⊢ ℲxB

Assertion

Ref	Expression
nfmpt	⊢ Ⅎx(y ∈ 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 5653	. 2 ⊢ (y ∈ A ↦ B) = {⟨y, z⟩ ∣ (y ∈ A ∧ z = B)}
2		nfmpt.1	. . . . 5 ⊢ ℲxA
3	2	nfcri 2484	. . . 4 ⊢ Ⅎx y ∈ A
4		nfmpt.2	. . . . 5 ⊢ ℲxB
5	4	nfeq2 2501	. . . 4 ⊢ Ⅎx z = B
6	3, 5	nfan 1824	. . 3 ⊢ Ⅎx(y ∈ A ∧ z = B)
7	6	nfopab 4628	. 2 ⊢ Ⅎx{⟨y, z⟩ ∣ (y ∈ A ∧ z = B)}
8	1, 7	nfcxfr 2487	1 ⊢ Ⅎx(y ∈ A ↦ B)

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  {copab 4623   ↦ cmpt 5652

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-opab 4624  df-mpt 5653

This theorem is referenced by: (None)