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Theorem rdglem1 8415
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 2733 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem3 8378 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))}
3 fveq2 6892 . . . . . . 7 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
4 reseq2 5977 . . . . . . . 8 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
54fveq2d 6896 . . . . . . 7 (𝑣 = 𝑤 → (𝐺‘(𝑔𝑣)) = (𝐺‘(𝑔𝑤)))
63, 5eqeq12d 2749 . . . . . 6 (𝑣 = 𝑤 → ((𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
76cbvralvw 3235 . . . . 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 3095 . . 3 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
109abbii 2803 . 2 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
112, 10eqtri 2761 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 2710  wral 3062  wrex 3071  cres 5679  Oncon0 6365   Fn wfn 6539  cfv 6544
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  rdgseg  8422
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