Theorem rdglem1 8217

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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}

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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem3 8180	. 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)))}
3		fveq2 6756	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑔‘𝑣) = (𝑔‘𝑤))
4		reseq2 5875	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑔 ↾ 𝑣) = (𝑔 ↾ 𝑤))
5	4	fveq2d 6760	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝐺‘(𝑔 ↾ 𝑣)) = (𝐺‘(𝑔 ↾ 𝑤)))
6	3, 5	eqeq12d 2754	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
7	6	cbvralvw 3372	. . . . 5 ⊢ (∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))
8	7	anbi2i 622	. . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
9	8	rexbii 3177	. . 3 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	9	abbii 2809	. 2 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}
11	2, 10	eqtri 2766	1 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}

Syntax hints:   ∧ wa 395   = wceq 1539  {cab 2715  ∀wral 3063  ∃wrex 3064   ↾ cres 5582  Oncon0 6251   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  rdgseg  8224