Theorem tfrlem6 3911

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 3261	. . . 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 3909	. . . . 5 |- (g e. A -> Fun g)
5		funrel 3529	. . . . 5 |- (Fun g -> Rel g)
6	4, 5	syl 10	. . . 4 |- (g e. A -> Rel g)
7	2, 6	mprgbir 1699	. . 3 |- Rel U.A
8		releq 3239	. . 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 955   e. wcel 957  {cab 1462  A.wral 1643  E.wrex 1644  U.cuni 2499  Oncon0 2944   |` cres 3168  Rel wrel 3171  Fun wfun 3172   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  tfrlem7 3912  zorn2lem4 4774

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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