Theorem tfrlem12 3913

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 3575	. . . . 5 |- (x = suc dom F -> (C Fn x <-> C Fn suc dom F))
2		raleq1 1783	. . . . 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 627	. . . 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 1873	. . 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 3057	. . 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 3911	. . . 4 |- (dom F e. On -> C Fn suc dom F)
10	6, 7, 8	tfrlem11 3912	. . . . 5 |- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))
11	10	r19.21aiv 1710	. . . 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 3599	. . . 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 3574	. . . . . 6 |- (f = C -> (f Fn x <-> C Fn x))
17		fveq1 3714	. . . . . . . 8 |- (f = C -> (f` y) = (C` y))
18		reseq1 3360	. . . . . . . . 9 |- (f = C -> (f |` y) = (C |` y))
19	18	fveq2d 3719	. . . . . . . 8 |- (f = C -> (G` (f |` y)) = (G` (C |` y)))
20	17, 19	eqeq12d 1486	. . . . . . 7 |- (f = C -> ((f` y) = (G` (f |` y)) <-> (C` y) = (G` (C |` y))))
21	20	ralbidv 1660	. . . . . 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 627	. . . . 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 1661	. . . 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 1896	. . 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 954   e. wcel 956  {cab 1461  A.wral 1642  E.wrex 1643  Vcvv 1807   u. cun 2041  {csn 2405  <.cop 2407  U.cuni 2498  Oncon0 2943  suc csuc 2945  dom cdm 3165   |` cres 3167   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  tfrlem13 3914

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-9 963  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-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  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-rab 1649  df-v 1808  df-sbc 1938  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-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  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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