Theorem tfrlem3 3913

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 1813	. . . . 5 |- g e. V
3		fneq1 3582	. . . . . . 7 |- (f = g -> (f Fn x <-> g Fn x))
4		fveq1 3723	. . . . . . . . 9 |- (f = g -> (f` y) = (g` y))
5		reseq1 3368	. . . . . . . . . 10 |- (f = g -> (f |` y) = (g |` y))
6	5	fveq2d 3728	. . . . . . . . 9 |- (f = g -> (G` (f |` y)) = (G` (g |` y)))
7	4, 6	eqeq12d 1489	. . . . . . . 8 |- (f = g -> ((f` y) = (G` (f |` y)) <-> (g` y) = (G` (g |` y))))
8	7	ralbidv 1663	. . . . . . 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 628	. . . . . 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 1664	. . . . 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 1897	. . . 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 3583	. . . . . 6 |- (x = z -> (g Fn x <-> g Fn z))
13		raleq1 1786	. . . . . 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 628	. . . . 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 1801	. . . 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 1574	. 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 1495	1 |- A = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  Oncon0 2948   |` cres 3172   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tfrlem4 3914  tfrlem5 3915  rdglem1 3937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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