Theorem nfmpt1 4219

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 4189	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
2		nfopab1 4195	. 2 ⊢ Ⅎ𝑥{⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
3	1, 2	nfcxfr 2389	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Ⅎwnfc 2379  {copab 4186   ↦ cmpt 4187

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-opab 4188  df-mpt 4189

This theorem is referenced by:  nffvmpt1  5701  fvmptss2  5774  fvmptssdm  5784  fvmptdf  5787  mpteqb  5790  fvmptf  5792  ralrnmpt  5841  rexrnmpt  5842  f1ompt  5850  f1mpt  5967  fliftfun  5992  dom2lem  7048  mapxpen  7138  mkvprop  7488  cc3  7624  nfcprod1  12299  ballotfilem7  13257  cnmpt11  15307  lgseisenlem2  16104