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Theorem nfrecs 6204
 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 6202 . 2 recs(𝐹) = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2281 . . . . 5 𝑥On
3 nfv 1508 . . . . . 6 𝑥 𝑎 Fn 𝑏
4 nfcv 2281 . . . . . . 7 𝑥𝑏
5 nfrecs.f . . . . . . . . 9 𝑥𝐹
6 nfcv 2281 . . . . . . . . 9 𝑥(𝑎𝑐)
75, 6nffv 5431 . . . . . . . 8 𝑥(𝐹‘(𝑎𝑐))
87nfeq2 2293 . . . . . . 7 𝑥(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2471 . . . . . 6 𝑥𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1544 . . . . 5 𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2472 . . . 4 𝑥𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2286 . . 3 𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3742 . 2 𝑥 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2278 1 𝑥recs(𝐹)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331  {cab 2125  Ⅎwnfc 2268  ∀wral 2416  ∃wrex 2417  ∪ cuni 3736  Oncon0 4285   ↾ cres 4541   Fn wfn 5118  ‘cfv 5123  recscrecs 6201 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-recs 6202 This theorem is referenced by:  nffrec  6293
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