Theorem rdglem1 3937

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 | 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.w e. z (g` w) = (G` (g |` w)))}

Distinct variable groups:   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		eqid 1475	. . 3 |- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
2	1	tfrlem3 3913	. 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)))}
3		fveq2 3724	. . . . . . 7 |- (y = w -> (g` y) = (g` w))
4		reseq2 3369	. . . . . . . 8 |- (y = w -> (g |` y) = (g |` w))
5	4	fveq2d 3728	. . . . . . 7 |- (y = w -> (G` (g |` y)) = (G` (g |` w)))
6	3, 5	eqeq12d 1489	. . . . . 6 |- (y = w -> ((g` y) = (G` (g |` y)) <-> (g` w) = (G` (g |` w))))
7	6	cbvralv 1800	. . . . 5 |- (A.y e. z (g` y) = (G` (g |` y)) <-> A.w e. z (g` w) = (G` (g |` w)))
8	7	anbi2i 480	. . . 4 |- ((g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) <-> (g Fn z /\ A.w e. z (g` w) = (G` (g |` w))))
9	8	rexbii 1668	. . 3 |- (E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) <-> E.z e. On (g Fn z /\ A.w e. z (g` w) = (G` (g |` w))))
10	9	abbii 1575	. 2 |- {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))} = {g | E.z e. On (g Fn z /\ A.w e. z (g` w) = (G` (g |` w)))}
11	2, 10	eqtr 1495	1 |- {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.w e. z (g` w) = (G` (g |` w)))}

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

This theorem is referenced by:  rdgfnon 3939  rdgval 3940  numth 4784  zorn2 4796

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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