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Theorem rdglem1 7685
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 2774 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem3 7648 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))}
3 fveq2 6348 . . . . . . 7 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
4 reseq2 5541 . . . . . . . 8 (𝑣 = 𝑤 → (𝑔𝑣) = (𝑔𝑤))
54fveq2d 6352 . . . . . . 7 (𝑣 = 𝑤 → (𝐺‘(𝑔𝑣)) = (𝐺‘(𝑔𝑤)))
63, 5eqeq12d 2789 . . . . . 6 (𝑣 = 𝑤 → ((𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
76cbvralv 3324 . . . . 5 (∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))
87anbi2i 610 . . . 4 ((𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
98rexbii 3193 . . 3 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
109abbii 2891 . 2 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣𝑧 (𝑔𝑣) = (𝐺‘(𝑔𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
112, 10eqtri 2796 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 1634  {cab 2760  wral 3064  wrex 3065  cres 5265  Oncon0 5877   Fn wfn 6037  cfv 6042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-res 5275  df-iota 6005  df-fun 6044  df-fn 6045  df-fv 6050
This theorem is referenced by:  rdgseg  7692
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