Theorem tfrlem3 3898

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 some bound variables in A for later use.

Hypothesis

Ref	Expression
tfrlem3.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}

Assertion

Ref	Expression
tfrlem3	|- A = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}

Distinct variable groups:   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 | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
2		visset 1804	. . . . 5 |- g e. V
3		fneq1 3568	. . . . . . 7 |- (f = g -> (f Fn x <-> g Fn x))
4		fveq1 3708	. . . . . . . . 9 |- (f = g -> (f` y) = (g` y))
5		reseq1 3352	. . . . . . . . . 10 |- (f = g -> (f |` y) = (g |` y))
6	5	fveq2d 3713	. . . . . . . . 9 |- (f = g -> (G` (f |` y)) = (G` (g |` y)))
7	4, 6	eqeq12d 1481	. . . . . . . 8 |- (f = g -> ((f` y) = (G` (f |` y)) <-> (g` y) = (G` (g |` y))))
8	7	ralbidv 1655	. . . . . . 7 |- (f = g -> (A.y e. x (f` y) = (G` (f |` y)) <-> A.y e. x (g` y) = (G` (g |` y))))
9	3, 8	anbi12d 626	. . . . . 6 |- (f = g -> ((f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> (g Fn x /\ A.y e. x (g` y) = (G` (g |` y)))))
10	9	rexbidv 1656	. . . . 5 |- (f = g -> (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> E.x e. On (g Fn x /\ A.y e. x (g` y) = (G` (g |` y)))))
11	2, 10	elab 1888	. . . 4 |- (g e. {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} <-> E.x e. On (g Fn x /\ A.y e. x (g` y) = (G` (g |` y))))
12		fneq2 3569	. . . . . 6 |- (x = z -> (g Fn x <-> g Fn z))
13		raleq1 1778	. . . . . 6 |- (x = z -> (A.y e. x (g` y) = (G` (g |` y)) <-> A.y e. z (g` y) = (G` (g |` y))))
14	12, 13	anbi12d 626	. . . . 5 |- (x = z -> ((g Fn x /\ A.y e. x (g` y) = (G` (g |` y))) <-> (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))))
15	14	cbvrexv 1792	. . . 4 |- (E.x e. On (g Fn x /\ A.y e. x (g` y) = (G` (g |` y))) <-> E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))))
16	11, 15	bitr 173	. . 3 |- (g e. {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} <-> E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))))
17	16	abbi2i 1566	. 2 |- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}
18	1, 17	eqtr 1487	1 |- A = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637  E.wrex 1638  Oncon0 2938   |` cres 3162   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tfrlem4 3899  tfrlem5 3900  rdglem1 3922

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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