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Theorem nfrecs 6158
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 6156 . 2 recs(𝐹) = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2255 . . . . 5 𝑥On
3 nfv 1491 . . . . . 6 𝑥 𝑎 Fn 𝑏
4 nfcv 2255 . . . . . . 7 𝑥𝑏
5 nfrecs.f . . . . . . . . 9 𝑥𝐹
6 nfcv 2255 . . . . . . . . 9 𝑥(𝑎𝑐)
75, 6nffv 5385 . . . . . . . 8 𝑥(𝐹‘(𝑎𝑐))
87nfeq2 2267 . . . . . . 7 𝑥(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2445 . . . . . 6 𝑥𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1527 . . . . 5 𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2446 . . . 4 𝑥𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2260 . . 3 𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3708 . 2 𝑥 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2252 1 𝑥recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  {cab 2101  wnfc 2242  wral 2390  wrex 2391   cuni 3702  Oncon0 4245  cres 4501   Fn wfn 5076  cfv 5081  recscrecs 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-recs 6156
This theorem is referenced by:  nffrec  6247
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