Theorem nfmpt1 4075

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)

Assertion

Ref	Expression
nfmpt1	⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)

Proof of Theorem nfmpt1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4045	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
2		nfopab1 4051	. 2 ⊢ Ⅎ𝑥{⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
3	1, 2	nfcxfr 2305	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Ⅎwnfc 2295  {copab 4042   ↦ cmpt 4043

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-opab 4044  df-mpt 4045

This theorem is referenced by:  nffvmpt1  5497  fvmptss2  5561  fvmptssdm  5570  fvmptdf  5573  mpteqb  5576  fvmptf  5578  ralrnmpt  5627  rexrnmpt  5628  f1ompt  5636  f1mpt  5739  fliftfun  5764  dom2lem  6738  mapxpen  6814  mkvprop  7122  cc3  7209  nfcprod1  11495  cnmpt11  12923