Theorem rdglem1 8430

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 2760	. . 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 8392	. 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 6890	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑔‘𝑣) = (𝑔‘𝑤))
4		reseq2 5966	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑔 ↾ 𝑣) = (𝑔 ↾ 𝑤))
5	4	fveq2d 6894	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝐺‘(𝑔 ↾ 𝑣)) = (𝐺‘(𝑔 ↾ 𝑤)))
6	3, 5	eqeq12d 2776	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
7	6	cbvralvw 3240	. . . . 5 ⊢ (∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))
8	7	anbi2i 635	. . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
9	8	rexbii 3109	. . 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 2827	. 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 2783	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)))}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  {cab 2738  ∀wral 3076  ∃wrex 3086   ↾ cres 5653  Oncon0 6365   Fn wfn 6538  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6497  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is used by:  rdgseg  8437