Theorem nfmpt1 4136

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 4106	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
2		nfopab1 4112	. 2 ⊢ Ⅎ𝑥{⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
3	1, 2	nfcxfr 2344	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1372   ∈ wcel 2175  Ⅎwnfc 2334  {copab 4103   ↦ cmpt 4104

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-opab 4105  df-mpt 4106

This theorem is referenced by:  nffvmpt1  5586  fvmptss2  5653  fvmptssdm  5663  fvmptdf  5666  mpteqb  5669  fvmptf  5671  ralrnmpt  5721  rexrnmpt  5722  f1ompt  5730  f1mpt  5839  fliftfun  5864  dom2lem  6862  mapxpen  6944  mkvprop  7259  cc3  7379  nfcprod1  11807  cnmpt11  14697  lgseisenlem2  15490