Theorem rdglem1 8043

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 2819	. . 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 8006	. 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 6663	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑔‘𝑣) = (𝑔‘𝑤))
4		reseq2 5841	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑔 ↾ 𝑣) = (𝑔 ↾ 𝑤))
5	4	fveq2d 6667	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝐺‘(𝑔 ↾ 𝑣)) = (𝐺‘(𝑔 ↾ 𝑤)))
6	3, 5	eqeq12d 2835	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
7	6	cbvralvw 3448	. . . . 5 ⊢ (∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))
8	7	anbi2i 624	. . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
9	8	rexbii 3245	. . 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 2884	. 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 2842	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 398   = wceq 1531  {cab 2797  ∀wral 3136  ∃wrex 3137   ↾ cres 5550  Oncon0 6184   Fn wfn 6343  ‘cfv 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356

This theorem is referenced by:  rdgseg  8050