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Theorem nffrecs 33017
Description: Bound-variable hypothesis builder for the founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.)
Hypotheses
Ref Expression
nffrecs.1 𝑥𝑅
nffrecs.2 𝑥𝐴
nffrecs.3 𝑥𝐹
Assertion
Ref Expression
nffrecs 𝑥frecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nffrecs
Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frecs 33015 . 2 frecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2 nfv 1906 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2974 . . . . . . . 8 𝑥𝑦
4 nffrecs.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3957 . . . . . . 7 𝑥 𝑦𝐴
6 nffrecs.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2974 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 6146 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3957 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfralw 3222 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1891 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nffrecs.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2974 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5848 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
157, 12, 14nfov 7175 . . . . . . . 8 𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2992 . . . . . . 7 𝑥(𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfralw 3222 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1893 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2334 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2981 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4837 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2972 1 𝑥frecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1079   = wceq 1528  wex 1771  {cab 2796  wnfc 2958  wral 3135  wss 3933   cuni 4830  cres 5550  Predcpred 6140   Fn wfn 6343  cfv 6348  (class class class)co 7145  frecscfrecs 33014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-ov 7148  df-frecs 33015
This theorem is referenced by: (None)
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