ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfrecs GIF version

Theorem nfrecs 6332
Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
nfrecs.f 𝑥𝐹
Assertion
Ref Expression
nfrecs 𝑥recs(𝐹)

Proof of Theorem nfrecs
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-recs 6330 . 2 recs(𝐹) = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2332 . . . . 5 𝑥On
3 nfv 1539 . . . . . 6 𝑥 𝑎 Fn 𝑏
4 nfcv 2332 . . . . . . 7 𝑥𝑏
5 nfrecs.f . . . . . . . . 9 𝑥𝐹
6 nfcv 2332 . . . . . . . . 9 𝑥(𝑎𝑐)
75, 6nffv 5544 . . . . . . . 8 𝑥(𝐹‘(𝑎𝑐))
87nfeq2 2344 . . . . . . 7 𝑥(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2528 . . . . . 6 𝑥𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1576 . . . . 5 𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2529 . . . 4 𝑥𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2337 . . 3 𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3830 . 2 𝑥 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2329 1 𝑥recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  {cab 2175  wnfc 2319  wral 2468  wrex 2469   cuni 3824  Oncon0 4381  cres 4646   Fn wfn 5230  cfv 5235  recscrecs 6329
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-recs 6330
This theorem is referenced by:  nffrec  6421
  Copyright terms: Public domain W3C validator