Theorem rdglem1 2975

Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.

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 group(s):   x,y,f,g   x,z,y,g   f,G,g,x   z,G   y,w,G,z,g

Proof of Theorem rdglem1

Step	Hyp	Ref	Expression
1		cleqid 1102	. . 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 2951	. 2 ⊢ {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))} = {g∣∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y)))}
3		fveq2 2832	. . . . . . 7 ⊢ (y = w → (g ‘y) = (g ‘w))
4		reseq2 2576	. . . . . . . 8 ⊢ (y = w → (g ↾ y) = (g ↾ w))
5	4	fveq2d 2836	. . . . . . 7 ⊢ (y = w → (G ‘(g ↾ y)) = (G ‘(g ↾ w)))
6	3, 5	cleq12d 1115	. . . . . 6 ⊢ (y = w → ((g ‘y) = (G ‘(g ↾ y)) ↔ (g ‘w) = (G ‘(g ↾ w))))
7	6	cbvralv 1333	. . . . 5 ⊢ (∀y ∈ z (g ‘y) = (G ‘(g ↾ y)) ↔ ∀w ∈ z (g ‘w) = (G ‘(g ↾ w)))
8	7	anbi2i 367	. . . 4 ⊢ ((g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y))) ↔ (g Fn z ∧ ∀w ∈ z (g ‘w) = (G ‘(g ↾ w))))
9	8	birex 1224	. . 3 ⊢ (∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y))) ↔ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g ‘w) = (G ‘(g ↾ w))))
10	9	biabi 1181	. 2 ⊢ {g∣∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y)))} = {g∣∃z ∈ On (g Fn z ∧ ∀w ∈ z (g ‘w) = (G ‘(g ↾ w)))}
11	2, 10	eqtr 1119	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 196   = weq 797  {cab 1090   = wceq 1091  ∀wral 1201  ∃wrex 1202  Oncon0 2199   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  rdgfnon 2977  rdgval 2978  numth 3599  zorn 3611

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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