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Theorem rdglem1 8454
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 2735 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem3 8417 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))}
3 fveq2 6907 . . . . . . 7 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
4 reseq2 5995 . . . . . . . 8 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
54fveq2d 6911 . . . . . . 7 (𝑣 = 𝑤 → (𝐺‘(𝑔𝑣)) = (𝐺‘(𝑔𝑤)))
63, 5eqeq12d 2751 . . . . . 6 (𝑣 = 𝑤 → ((𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
76cbvralvw 3235 . . . . 5 (∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))
87anbi2i 623 . . . 4 ((𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
98rexbii 3092 . . 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 2763 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 395   = wceq 1537  {cab 2712  wral 3059  wrex 3068  cres 5691  Oncon0 6386   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  rdgseg  8461
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