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

Theorem nfrecs 6283
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 6281 . 2 recs(𝐹) = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2312 . . . . 5 𝑥On
3 nfv 1521 . . . . . 6 𝑥 𝑎 Fn 𝑏
4 nfcv 2312 . . . . . . 7 𝑥𝑏
5 nfrecs.f . . . . . . . . 9 𝑥𝐹
6 nfcv 2312 . . . . . . . . 9 𝑥(𝑎𝑐)
75, 6nffv 5504 . . . . . . . 8 𝑥(𝐹‘(𝑎𝑐))
87nfeq2 2324 . . . . . . 7 𝑥(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2508 . . . . . 6 𝑥𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1558 . . . . 5 𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2509 . . . 4 𝑥𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2317 . . 3 𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3800 . 2 𝑥 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2309 1 𝑥recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  {cab 2156  wnfc 2299  wral 2448  wrex 2449   cuni 3794  Oncon0 4346  cres 4611   Fn wfn 5191  cfv 5196  recscrecs 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-recs 6281
This theorem is referenced by:  nffrec  6372
  Copyright terms: Public domain W3C validator