MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdglem1 Structured version   Visualization version   GIF version

Theorem rdglem1 7398
Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)
Assertion
Ref Expression
rdglem1 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
Distinct variable groups:   𝑥,𝑦,𝑓,𝑔,𝑧,𝐺   𝑦,𝑤,𝐺,𝑧,𝑔

Proof of Theorem rdglem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem3 7361 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))}
3 fveq2 6103 . . . . . . 7 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
4 reseq2 5312 . . . . . . . 8 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
54fveq2d 6107 . . . . . . 7 (𝑣 = 𝑤 → (𝐺‘(𝑔𝑣)) = (𝐺‘(𝑔𝑤)))
63, 5eqeq12d 2625 . . . . . 6 (𝑣 = 𝑤 → ((𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
76cbvralv 3147 . . . . 5 (∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))
87anbi2i 726 . . . 4 ((𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
98rexbii 3023 . . 3 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
109abbii 2726 . 2 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
112, 10eqtri 2632 1 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  {cab 2596  wral 2896  wrex 2897  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:  rdgseg  7405
  Copyright terms: Public domain W3C validator