MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfrlem3a Structured version   Visualization version   GIF version

Theorem wfrlem3a 7946
Description: Lemma for well-founded recursion. Show membership in the class of acceptable functions. (Contributed by Scott Fenton, 31-Jul-2020.)
Hypotheses
Ref Expression
wfrlem1.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
wfrlem3a.2 𝐺 ∈ V
Assertion
Ref Expression
wfrlem3a (𝐺𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
Distinct variable groups:   𝐴,𝑓,𝑤,𝑥,𝑦,𝑧   𝑓,𝐹,𝑤,𝑥,𝑦,𝑧   𝑅,𝑓,𝑤,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓)

Proof of Theorem wfrlem3a
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 wfrlem3a.2 . 2 𝐺 ∈ V
2 fneq1 6437 . . . 4 (𝑔 = 𝐺 → (𝑔 Fn 𝑧𝐺 Fn 𝑧))
3 fveq1 6662 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑤) = (𝐺𝑤))
4 reseq1 5840 . . . . . . 7 (𝑔 = 𝐺 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))
54fveq2d 6667 . . . . . 6 (𝑔 = 𝐺 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))
63, 5eqeq12d 2834 . . . . 5 (𝑔 = 𝐺 → ((𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
76ralbidv 3194 . . . 4 (𝑔 = 𝐺 → (∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
82, 73anbi13d 1429 . . 3 (𝑔 = 𝐺 → ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐺 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))))
98exbidv 1913 . 2 (𝑔 = 𝐺 → (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))))
10 wfrlem1.1 . . 3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1110wfrlem1 7943 . 2 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
121, 9, 11elab2 3667 1 (𝐺𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  Vcvv 3492  wss 3933  cres 5550  Predcpred 6140   Fn wfn 6343  cfv 6348
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-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  wfrlem17  7960
  Copyright terms: Public domain W3C validator