MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem3a Structured version   Visualization version   GIF version

Theorem tfrlem3a 7360
Description: Lemma for transfinite recursion. Let 𝐴 be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in 𝐴 for later use. (Contributed by NM, 9-Apr-1995.)
Hypotheses
Ref Expression
tfrlem3.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem3.2 𝐺 ∈ V
Assertion
Ref Expression
tfrlem3a (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
Distinct variable groups:   𝑤,𝑓,𝑥,𝑦,𝑧,𝐹   𝑓,𝐺,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑓)

Proof of Theorem tfrlem3a
StepHypRef Expression
1 tfrlem3.2 . 2 𝐺 ∈ V
2 fneq12 5898 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (𝑓 Fn 𝑥𝐺 Fn 𝑧))
3 simpll 786 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4 simpr 476 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
53, 4fveq12d 6109 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
63, 4reseq12d 5318 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
76fveq2d 6107 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐺𝑤)))
85, 7eqeq12d 2625 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
9 simplr 788 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
108, 9cbvraldva2 3151 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
112, 10anbi12d 743 . . 3 ((𝑓 = 𝐺𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
1211cbvrexdva 3154 . 2 (𝑓 = 𝐺 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
13 tfrlem3.1 . 2 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
141, 12, 13elab2 3323 1 (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  cres 5040  Oncon0 5640   Fn wfn 5799  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812
This theorem is referenced by:  tfrlem3  7361  tfrlem5  7363  tfrlem9a  7369
  Copyright terms: Public domain W3C validator