Theorem tfrlem6 3916

Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation.

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A

Assertion

Ref	Expression
tfrlem6	|- Rel F

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfrlem6

Step	Hyp	Ref	Expression
1		tfrlem.2	. 2 |- F = U.A
2		reluni 3265	. . . 4 |- (Rel U.A <-> A.g e. A Rel g)
3		tfrlem.1	. . . . . 6 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
4	3, 1	tfrlem4 3914	. . . . 5 |- (g e. A -> Fun g)
5		funrel 3533	. . . . 5 |- (Fun g -> Rel g)
6	4, 5	syl 10	. . . 4 |- (g e. A -> Rel g)
7	2, 6	mprgbir 1701	. . 3 |- Rel U.A
8		releq 3243	. . 3 |- (F = U.A -> (Rel F <-> Rel U.A))
9	7, 8	mpbiri 194	. 2 |- (F = U.A -> Rel F)
10	1, 9	ax-mp 7	1 |- Rel F

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  U.cuni 2503  Oncon0 2948   |` cres 3172  Rel wrel 3175  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tfrlem7 3917  zorn2lem4 4791

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

Copyright terms: Public domain