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Theorem nfrecs 5863
 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 5861 . 2 recs(𝐹) = {𝑎𝑏 On (𝑎 Fn 𝑏 𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
2 nfcv 2175 . . . . 5 xOn
3 nfv 1418 . . . . . 6 x 𝑎 Fn 𝑏
4 nfcv 2175 . . . . . . 7 x𝑏
5 nfrecs.f . . . . . . . . 9 x𝐹
6 nfcv 2175 . . . . . . . . 9 x(𝑎𝑐)
75, 6nffv 5128 . . . . . . . 8 x(𝐹‘(𝑎𝑐))
87nfeq2 2186 . . . . . . 7 x(𝑎𝑐) = (𝐹‘(𝑎𝑐))
94, 8nfralxy 2354 . . . . . 6 x𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐))
103, 9nfan 1454 . . . . 5 x(𝑎 Fn 𝑏 𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
112, 10nfrexxy 2355 . . . 4 x𝑏 On (𝑎 Fn 𝑏 𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))
1211nfab 2179 . . 3 x{𝑎𝑏 On (𝑎 Fn 𝑏 𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
1312nfuni 3577 . 2 x {𝑎𝑏 On (𝑎 Fn 𝑏 𝑐 𝑏 (𝑎𝑐) = (𝐹‘(𝑎𝑐)))}
141, 13nfcxfr 2172 1 xrecs(𝐹)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  {cab 2023  Ⅎwnfc 2162  ∀wral 2300  ∃wrex 2301  ∪ cuni 3571  Oncon0 4066   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845  recscrecs 5860 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-recs 5861 This theorem is referenced by:  nffrec  5921
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