MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nffrecs Structured version   Visualization version   GIF version

Theorem nffrecs 8218
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 8216 . 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 3940 . . . . . . 7 𝑥 𝑦𝐴
6 nffrecs.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2904 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 6262 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3940 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfralw 3293 . . . . . . 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 5943 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
157, 12, 14nfov 7391 . . . . . . . 8 𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2921 . . . . . . 7 𝑥(𝑓𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfralw 3293 . . . . . 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 4876 . 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 3061  wss 3914   cuni 4869  cres 5639  Predcpred 6256   Fn wfn 6495  cfv 6500  (class class class)co 7361  frecscfrecs 8215
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fv 6508  df-ov 7364  df-frecs 8216
This theorem is referenced by:  nfwrecs  8251
  Copyright terms: Public domain W3C validator