Theorem rdglem1 3932

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 1474	. . 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 3908	. 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 3719	. . . . . . 7 |- (y = w -> (g` y) = (g` w))
4		reseq2 3365	. . . . . . . 8 |- (y = w -> (g |` y) = (g |` w))
5	4	fveq2d 3723	. . . . . . 7 |- (y = w -> (G` (g |` y)) = (G` (g |` w)))
6	3, 5	eqeq12d 1487	. . . . . 6 |- (y = w -> ((g` y) = (G` (g |` y)) <-> (g` w) = (G` (g |` w))))
7	6	cbvralv 1797	. . . . 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 1666	. . 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 1573	. 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 1493	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 955  {cab 1462  A.wral 1643  E.wrex 1644  Oncon0 2944   |` cres 3168   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  rdgfnon 3934  rdgval 3935  numth 4767  zorn2 4779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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