Theorem tfrlem3 2951

Description: Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes the bound variables in A for later use.

Hypothesis

Ref	Expression
tfrlem3.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}

Assertion

Ref	Expression
tfrlem3	⊢ A = {g∣∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y)))}

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

Proof of Theorem tfrlem3

Step	Hyp	Ref	Expression
1		tfrlem3.1	. 2 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
2		visset 1350	. . . . 5 ⊢ g ∈ V
3		fneq1 2718	. . . . . . 7 ⊢ (f = g → (f Fn x ↔ g Fn x))
4		fveq1 2831	. . . . . . . . 9 ⊢ (f = g → (f ‘y) = (g ‘y))
5		reseq1 2575	. . . . . . . . . 10 ⊢ (f = g → (f ↾ y) = (g ↾ y))
6	5	fveq2d 2836	. . . . . . . . 9 ⊢ (f = g → (G ‘(f ↾ y)) = (G ‘(g ↾ y)))
7	4, 6	cleq12d 1115	. . . . . . . 8 ⊢ (f = g → ((f ‘y) = (G ‘(f ↾ y)) ↔ (g ‘y) = (G ‘(g ↾ y))))
8	7	biraldv 1219	. . . . . . 7 ⊢ (f = g → (∀y ∈ x (f ‘y) = (G ‘(f ↾ y)) ↔ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y))))
9	3, 8	anbi12d 476	. . . . . 6 ⊢ (f = g → ((f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) ↔ (g Fn x ∧ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y)))))
10	9	birexdv 1220	. . . . 5 ⊢ (f = g → (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) ↔ ∃x ∈ On (g Fn x ∧ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y)))))
11	2, 10	elab 1415	. . . 4 ⊢ (g ∈ {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))} ↔ ∃x ∈ On (g Fn x ∧ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y))))
12		fneq2 2719	. . . . . 6 ⊢ (x = z → (g Fn x ↔ g Fn z))
13		raleq 1324	. . . . . 6 ⊢ (x = z → (∀y ∈ x (g ‘y) = (G ‘(g ↾ y)) ↔ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y))))
14	12, 13	anbi12d 476	. . . . 5 ⊢ (x = z → ((g Fn x ∧ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y))) ↔ (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y)))))
15	14	cbvrexv 1334	. . . 4 ⊢ (∃x ∈ On (g Fn x ∧ ∀y ∈ x (g ‘y) = (G ‘(g ↾ y))) ↔ ∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y))))
16	11, 15	bitr 151	. . 3 ⊢ (g ∈ {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))} ↔ ∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y))))
17	16	biabri 1180	. 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)))}
18	1, 17	eqtr 1119	1 ⊢ A = {g∣∃z ∈ On (g Fn z ∧ ∀y ∈ z (g ‘y) = (G ‘(g ↾ y)))}

Syntax hints:   ∧ wa 196   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Oncon0 2199   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem4 2952  tfrlem5 2953  rdglem1 2975

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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