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Theorem nfwrecs 7644
Description: Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)
Hypotheses
Ref Expression
nfwrecs.1 𝑥𝑅
nfwrecs.2 𝑥𝐴
nfwrecs.3 𝑥𝐹
Assertion
Ref Expression
nfwrecs 𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecs
Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wrecs 7642 . 2 wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2 nfv 2005 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2948 . . . . . . . 8 𝑥𝑦
4 nfwrecs.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3791 . . . . . . 7 𝑥 𝑦𝐴
6 nfwrecs.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2948 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 5898 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3791 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfral 3133 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1990 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nfwrecs.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2948 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5599 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
1512, 14nffv 6418 . . . . . . . 8 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2964 . . . . . . 7 𝑥(𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfral 3133 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1993 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2330 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2953 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4636 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2946 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1100   = wceq 1637  wex 1859  {cab 2792  wnfc 2935  wral 3096  wss 3769   cuni 4630  cres 5313  Predcpred 5892   Fn wfn 6096  cfv 6101  wrecscwrecs 7641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-xp 5317  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-iota 6064  df-fv 6109  df-wrecs 7642
This theorem is referenced by:  nfrecs  7707
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