Theorem nfmpt1 4091

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 4061	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
2		nfopab1 4067	. 2 ⊢ Ⅎ𝑥{⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
3	1, 2	nfcxfr 2314	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2146  Ⅎwnfc 2304  {copab 4058   ↦ cmpt 4059

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-opab 4060  df-mpt 4061

This theorem is referenced by:  nffvmpt1  5518  fvmptss2  5583  fvmptssdm  5592  fvmptdf  5595  mpteqb  5598  fvmptf  5600  ralrnmpt  5650  rexrnmpt  5651  f1ompt  5659  f1mpt  5762  fliftfun  5787  dom2lem  6762  mapxpen  6838  mkvprop  7146  cc3  7242  nfcprod1  11528  cnmpt11  13334