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Theorem nffrecs 32105
 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 32103 . 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 1992 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2902 . . . . . . . 8 𝑥𝑦
4 nffrecs.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3737 . . . . . . 7 𝑥 𝑦𝐴
6 nffrecs.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2902 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 5846 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3737 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfral 3083 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1977 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nffrecs.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2902 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5553 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
157, 12, 14nfov 6840 . . . . . . . 8 𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2918 . . . . . . 7 𝑥(𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfral 3083 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1980 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2301 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2907 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4594 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2900 1 𝑥frecs(𝑅, 𝐴, 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853  {cab 2746  Ⅎwnfc 2889  ∀wral 3050   ⊆ wss 3715  ∪ cuni 4588   ↾ cres 5268  Predcpred 5840   Fn wfn 6044  ‘cfv 6049  (class class class)co 6814  frecscfrecs 32102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-iota 6012  df-fv 6057  df-ov 6817  df-frecs 32103 This theorem is referenced by: (None)
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