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Theorem nffrecs 8268
Description: Bound-variable hypothesis builder for the well-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 8266 . 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 1918 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2904 . . . . . . . 8 𝑥𝑦
4 nffrecs.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3975 . . . . . . 7 𝑥 𝑦𝐴
6 nffrecs.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2904 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 6306 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3975 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfralw 3309 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1903 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nffrecs.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2904 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5984 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
157, 12, 14nfov 7439 . . . . . . . 8 𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2921 . . . . . . 7 𝑥(𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfralw 3309 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1905 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2318 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2910 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4916 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2902 1 𝑥frecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  wex 1782  {cab 2710  wnfc 2884  wral 3062  wss 3949   cuni 4909  cres 5679  Predcpred 6300   Fn wfn 6539  cfv 6544  (class class class)co 7409  frecscfrecs 8265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fv 6552  df-ov 7412  df-frecs 8266
This theorem is referenced by:  nfwrecs  8301
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