Theorem tfrlem12 3922

Description: Lemma for transfinite recursion. Show C is an acceptable 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
tfrlem.3	|- C = (F u. {<.dom F, (G` (F |` dom F))>.})

Assertion

Ref	Expression
tfrlem12	|- (dom F e. On -> C e. A)

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

Proof of Theorem tfrlem12

Step	Hyp	Ref	Expression
1		fneq2 3583	. . . . 5 |- (x = suc dom F -> (C Fn x <-> C Fn suc dom F))
2		raleq1 1786	. . . . 5 |- (x = suc dom F -> (A.y e. x (C` y) = (G` (C |` y)) <-> A.y e. suc dom F(C` y) = (G` (C |` y))))
3	1, 2	anbi12d 628	. . . 4 |- (x = suc dom F -> ((C Fn x /\ A.y e. x (C` y) = (G` (C |` y))) <-> (C Fn suc dom F /\ A.y e. suc dom F(C` y) = (G` (C |` y)))))
4	3	rcla4ev 1877	. . 3 |- ((suc dom F e. On /\ (C Fn suc dom F /\ A.y e. suc dom F(C` y) = (G` (C |` y)))) -> E.x e. On (C Fn x /\ A.y e. x (C` y) = (G` (C |` y))))
5		suceloni 3062	. . 3 |- (dom F e. On -> suc dom F e. On)
6		tfrlem.1	. . . . 5 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
7		tfrlem.2	. . . . 5 |- F = U.A
8		tfrlem.3	. . . . 5 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
9	6, 7, 8	tfrlem10 3920	. . . 4 |- (dom F e. On -> C Fn suc dom F)
10	6, 7, 8	tfrlem11 3921	. . . . 5 |- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))
11	10	r19.21aiv 1713	. . . 4 |- (dom F e. On -> A.y e. suc dom F(C` y) = (G` (C |` y)))
12	9, 11	jca 288	. . 3 |- (dom F e. On -> (C Fn suc dom F /\ A.y e. suc dom F(C` y) = (G` (C |` y))))
13	4, 5, 12	sylanc 471	. 2 |- (dom F e. On -> E.x e. On (C Fn x /\ A.y e. x (C` y) = (G` (C |` y))))
14		fnex 3607	. . . 4 |- ((C Fn suc dom F /\ suc dom F e. On) -> C e. V)
15	14, 9, 5	sylanc 471	. . 3 |- (dom F e. On -> C e. V)
16		fneq1 3582	. . . . . 6 |- (f = C -> (f Fn x <-> C Fn x))
17		fveq1 3723	. . . . . . . 8 |- (f = C -> (f` y) = (C` y))
18		reseq1 3368	. . . . . . . . 9 |- (f = C -> (f |` y) = (C |` y))
19	18	fveq2d 3728	. . . . . . . 8 |- (f = C -> (G` (f |` y)) = (G` (C |` y)))
20	17, 19	eqeq12d 1489	. . . . . . 7 |- (f = C -> ((f` y) = (G` (f |` y)) <-> (C` y) = (G` (C |` y))))
21	20	ralbidv 1663	. . . . . 6 |- (f = C -> (A.y e. x (f` y) = (G` (f |` y)) <-> A.y e. x (C` y) = (G` (C |` y))))
22	16, 21	anbi12d 628	. . . . 5 |- (f = C -> ((f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> (C Fn x /\ A.y e. x (C` y) = (G` (C |` y)))))
23	22	rexbidv 1664	. . . 4 |- (f = C -> (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> E.x e. On (C Fn x /\ A.y e. x (C` y) = (G` (C |` y)))))
24	23, 6	elab2g 1900	. . 3 |- (C e. V -> (C e. A <-> E.x e. On (C Fn x /\ A.y e. x (C` y) = (G` (C |` y)))))
25	15, 24	syl 10	. 2 |- (dom F e. On -> (C e. A <-> E.x e. On (C Fn x /\ A.y e. x (C` y) = (G` (C |` y)))))
26	13, 25	mpbird 196	1 |- (dom F e. On -> C e. A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  Vcvv 1811   u. cun 2045  {csn 2409  <.cop 2411  U.cuni 2503  Oncon0 2948  suc csuc 2950  dom cdm 3170   |` cres 3172   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tfrlem13 3923

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  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-rab 1652  df-v 1812  df-sbc 1942  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-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  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

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