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

Theorem nfrecs 5953
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 5951 . 2 recs(𝐹) = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2194 . . . . 5 𝑥On
3 nfv 1437 . . . . . 6 𝑥 𝑎 Fn 𝑏
4 nfcv 2194 . . . . . . 7 𝑥𝑏
5 nfrecs.f . . . . . . . . 9 𝑥𝐹
6 nfcv 2194 . . . . . . . . 9 𝑥(𝑎𝑐)
75, 6nffv 5213 . . . . . . . 8 𝑥(𝐹‘(𝑎𝑐))
87nfeq2 2205 . . . . . . 7 𝑥(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2377 . . . . . 6 𝑥𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1473 . . . . 5 𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2378 . . . 4 𝑥𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2198 . . 3 𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3614 . 2 𝑥 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2191 1 𝑥recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  {cab 2042  wnfc 2181  wral 2323  wrex 2324   cuni 3608  Oncon0 4128  cres 4375   Fn wfn 4925  cfv 4930  recscrecs 5950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-recs 5951
This theorem is referenced by:  nffrec  6013
  Copyright terms: Public domain W3C validator