Theorem tfrlem11 3921

Description: Lemma for transfinite recursion. Compute the value of C.

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
tfrlem11	|- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))

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

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		ssun1 2193	. . . . . . . . 9 |- F (_ (F u. {<.dom F, (G` (F |` dom F))>.})
2		tfrlem.3	. . . . . . . . 9 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
3	1, 2	sseqtr4 2094	. . . . . . . 8 |- F (_ C
4		funssfv 3735	. . . . . . . . . . . 12 |- ((Fun C /\ F (_ C /\ y e. dom F) -> (C` y) = (F` y))
5	4	3expa 833	. . . . . . . . . . 11 |- (((Fun C /\ F (_ C) /\ y e. dom F) -> (C` y) = (F` y))
6	5	adantrl 394	. . . . . . . . . 10 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (C` y) = (F` y))
7		fun2ssres 3553	. . . . . . . . . . . . 13 |- ((Fun C /\ F (_ C /\ y (_ dom F) -> (C |` y) = (F |` y))
8	7	3expa 833	. . . . . . . . . . . 12 |- (((Fun C /\ F (_ C) /\ y (_ dom F) -> (C |` y) = (F |` y))
9	8	fveq2d 3728	. . . . . . . . . . 11 |- (((Fun C /\ F (_ C) /\ y (_ dom F) -> (G` (C |` y)) = (G` (F |` y)))
10		onelsst 3000	. . . . . . . . . . . 12 |- (dom F e. On -> (y e. dom F -> y (_ dom F))
11	10	imp 350	. . . . . . . . . . 11 |- ((dom F e. On /\ y e. dom F) -> y (_ dom F)
12	9, 11	sylan2 451	. . . . . . . . . 10 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (G` (C |` y)) = (G` (F |` y)))
13	6, 12	eqeq12d 1489	. . . . . . . . 9 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> ((C` y) = (G` (C |` y)) <-> (F` y) = (G` (F |` y))))
14		tfrlem.1	. . . . . . . . . 10 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
15		tfrlem.2	. . . . . . . . . 10 |- F = U.A
16	14, 15	tfrlem9 3919	. . . . . . . . 9 |- (y e. dom F -> (F` y) = (G` (F |` y)))
17	13, 16	syl5bir 210	. . . . . . . 8 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
18	3, 17	mpanl2 707	. . . . . . 7 |- ((Fun C /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
19	14, 15, 2	tfrlem10 3920	. . . . . . . 8 |- (dom F e. On -> C Fn suc dom F)
20		fnfun 3585	. . . . . . . 8 |- (C Fn suc dom F -> Fun C)
21	19, 20	syl 10	. . . . . . 7 |- (dom F e. On -> Fun C)
22	18, 21	sylan 448	. . . . . 6 |- ((dom F e. On /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
23	22	exp32 377	. . . . 5 |- (dom F e. On -> (dom F e. On -> (y e. dom F -> (y e. dom F -> (C` y) = (G` (C |` y))))))
24	23	pm2.43i 64	. . . 4 |- (dom F e. On -> (y e. dom F -> (y e. dom F -> (C` y) = (G` (C |` y)))))
25	24	pm2.43d 65	. . 3 |- (dom F e. On -> (y e. dom F -> (C` y) = (G` (C |` y))))
26		opeq1 2487	. . . . . . . . . . 11 |- (y = dom F -> <.y, (G` (C |` y))>. = <.dom F, (G` (C |` y))>.)
27	26	adantl 388	. . . . . . . . . 10 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. = <.dom F, (G` (C |` y))>.)
28	3, 7	mp3an2 904	. . . . . . . . . . . . 13 |- ((Fun C /\ y (_ dom F) -> (C |` y) = (F |` y))
29		eqimss 2109	. . . . . . . . . . . . 13 |- (y = dom F -> y (_ dom F)
30	28, 21, 29	syl2an 454	. . . . . . . . . . . 12 |- ((dom F e. On /\ y = dom F) -> (C |` y) = (F |` y))
31		reseq2 3369	. . . . . . . . . . . . 13 |- (y = dom F -> (F |` y) = (F |` dom F))
32	31	adantl 388	. . . . . . . . . . . 12 |- ((dom F e. On /\ y = dom F) -> (F |` y) = (F |` dom F))
33	30, 32	eqtrd 1507	. . . . . . . . . . 11 |- ((dom F e. On /\ y = dom F) -> (C |` y) = (F |` dom F))
34		fveq2 3724	. . . . . . . . . . 11 |- ((C |` y) = (F |` dom F) -> (G` (C |` y)) = (G` (F |` dom F)))
35		opeq2 2488	. . . . . . . . . . 11 |- ((G` (C |` y)) = (G` (F |` dom F)) -> <.dom F, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
36	33, 34, 35	3syl 20	. . . . . . . . . 10 |- ((dom F e. On /\ y = dom F) -> <.dom F, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
37	27, 36	eqtrd 1507	. . . . . . . . 9 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
38	37	sneqd 2419	. . . . . . . 8 |- ((dom F e. On /\ y = dom F) -> {<.y, (G` (C |` y))>.} = {<.dom F, (G` (F |` dom F))>.})
39		opex 2782	. . . . . . . . 9 |- <.y, (G` (C |` y))>. e. V
40	39	snid 2435	. . . . . . . 8 |- <.y, (G` (C |` y))>. e. {<.y, (G` (C |` y))>.}
41	38, 40	syl5eleq 1554	. . . . . . 7 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. {<.dom F, (G` (F |` dom F))>.})
42		elun2 2198	. . . . . . 7 |- (<.y, (G` (C |` y))>. e. {<.dom F, (G` (F |` dom F))>.} -> <.y, (G` (C |` y))>. e. (F u. {<.dom F, (G` (F |` dom F))>.}))
43	41, 42	syl 10	. . . . . 6 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. (F u. {<.dom F, (G` (F |` dom F))>.}))
44	43, 2	syl6eleqr 1559	. . . . 5 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. C)
45		fvex 3732	. . . . . . 7 |- (G` (C |` y)) e. V
46	45	fnopfvb 3754	. . . . . 6 |- ((C Fn suc dom F /\ y e. suc dom F) -> ((C` y) = (G` (C |` y)) <-> <.y, (G` (C |` y))>. e. C))
47		visset 1813	. . . . . . 7 |- y e. V
48	47	eqelsuc 3054	. . . . . 6 |- (y = dom F -> y e. suc dom F)
49	46, 19, 48	syl2an 454	. . . . 5 |- ((dom F e. On /\ y = dom F) -> ((C` y) = (G` (C |` y)) <-> <.y, (G` (C |` y))>. e. C))
50	44, 49	mpbird 196	. . . 4 |- ((dom F e. On /\ y = dom F) -> (C` y) = (G` (C |` y)))
51	50	ex 373	. . 3 |- (dom F e. On -> (y = dom F -> (C` y) = (G` (C |` y))))
52	25, 51	jaod 424	. 2 |- (dom F e. On -> ((y e. dom F \/ y = dom F) -> (C` y) = (G` (C |` y))))
53		elsuci 3035	. 2 |- (y e. suc dom F -> (y e. dom F \/ y = dom F))
54	52, 53	syl5 21	1 |- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646   u. cun 2045   (_ wss 2047  {csn 2409  <.cop 2411  U.cuni 2503  Oncon0 2948  suc csuc 2950  dom cdm 3170   |` cres 3172  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tfrlem12 3922

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-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

Copyright terms: Public domain