Theorem tfrlem4 2952

Description: Lemma for transfinite recursion. A is the class of all "acceptable" functions, and F is their union. First we show that an acceptable function is in fact a function.

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
tfrlem4	|- (g e. A -> Fun g)

Distinct variable group(s):   x,y,f,g,A   x,F,y,f   x,G,y,f,g

Proof of Theorem tfrlem4

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
2	1	tfrlem3 2951	. . 3 |- A = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}
3	2	cleqabi 1176	. 2 |- (g e. A <-> E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))))
4		fnfun 2721	. . . . 5 |- (g Fn z -> Fun g)
5	4	adantr 306	. . . 4 |- ((g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g)
6	5	a1i 7	. . 3 |- (z e. On -> ((g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g))
7	6	r19.23aiv 1284	. 2 |- (E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g)
8	3, 7	sylbi 174	1 |- (g e. A -> Fun g)

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  U.cuni 1919  Oncon0 2199   |` cres 2412  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  tfrlem6 2954

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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