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Theorem rdglem1 8362
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 2737 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem3 8325 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))}
3 fveq2 6843 . . . . . . 7 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
4 reseq2 5933 . . . . . . . 8 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
54fveq2d 6847 . . . . . . 7 (𝑣 = 𝑤 → (𝐺‘(𝑔𝑣)) = (𝐺‘(𝑔𝑤)))
63, 5eqeq12d 2753 . . . . . 6 (𝑣 = 𝑤 → ((𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
76cbvralvw 3226 . . . . 5 (∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))
87anbi2i 624 . . . 4 ((𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
98rexbii 3098 . . 3 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
109abbii 2807 . 2 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
112, 10eqtri 2765 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 397   = wceq 1542  {cab 2714  wral 3065  wrex 3074  cres 5636  Oncon0 6318   Fn wfn 6492  cfv 6497
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-ext 2708
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  rdgseg  8369
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