Theorem tfrlem4 3905

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 groups:   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 3904	. . 3 |- A = {g | E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y)))}
3	2	abeq2i 1567	. 2 |- (g e. A <-> E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))))
4		fnfun 3577	. . . . 5 |- (g Fn z -> Fun g)
5	4	adantr 389	. . . 4 |- ((g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g)
6	5	a1i 8	. . 3 |- (z e. On -> ((g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g))
7	6	r19.23aiv 1740	. 2 |- (E.z e. On (g Fn z /\ A.y e. z (g` y) = (G` (g |` y))) -> Fun g)
8	3, 7	sylbi 199	1 |- (g e. A -> Fun g)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642  E.wrex 1643  U.cuni 2498  Oncon0 2943   |` cres 3167  Fun wfun 3171   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  tfrlem6 3907

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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