Theorem onfununi 5116

Description: A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
onfununi.1	|- (Lim y -> (F` y) = U_x e. y (F` x))
onfununi.2	|- ((x e. On /\ y e. On /\ x C_ y) -> (F` x) C_ (F` y))

Assertion

Ref	Expression
onfununi	|- ((S e. T /\ S C_ On /\ S =/= (/)) -> (F` U.S) = U_x e. S (F` x))

Distinct variable groups:   x,y,S   x,F,y   x,T

Proof of Theorem onfununi

Step	Hyp	Ref	Expression
1		ssorduni 3870	. . . . . . . . . . 11 |- (S C_ On -> Ord U.S)
2	1	ad2antrr 440	. . . . . . . . . 10 |- (((S C_ On /\ -. U.S e. S) /\ S =/= (/)) -> Ord U.S)
3		ssn0 2905	. . . . . . . . . . 11 |- ((S C_ U.S /\ S =/= (/)) -> U.S =/= (/))
4		elssuni 3206	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. S -> x C_ U.S)
5	4	adantl 424	. . . . . . . . . . . . . . . . . . . 20 |- ((S C_ On /\ x e. S) -> x C_ U.S)
6		ordsseleq 3687	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Ord x /\ Ord U.S) -> (x C_ U.S <-> (x e. U.S \/ x = U.S)))
7		ssel 2615	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (S C_ On -> (x e. S -> x e. On))
8		eloni 3667	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. On -> Ord x)
9	7, 8	syl6 25	. . . . . . . . . . . . . . . . . . . . . . 23 |- (S C_ On -> (x e. S -> Ord x))
10	9	imp 377	. . . . . . . . . . . . . . . . . . . . . 22 |- ((S C_ On /\ x e. S) -> Ord x)
11	6, 10, 1	syl2an 503	. . . . . . . . . . . . . . . . . . . . 21 |- (((S C_ On /\ x e. S) /\ S C_ On) -> (x C_ U.S <-> (x e. U.S \/ x = U.S)))
12	11	anabss1 557	. . . . . . . . . . . . . . . . . . . 20 |- ((S C_ On /\ x e. S) -> (x C_ U.S <-> (x e. U.S \/ x = U.S)))
13	5, 12	mpbid 212	. . . . . . . . . . . . . . . . . . 19 |- ((S C_ On /\ x e. S) -> (x e. U.S \/ x = U.S))
14	13	ord 249	. . . . . . . . . . . . . . . . . 18 |- ((S C_ On /\ x e. S) -> (-. x e. U.S -> x = U.S))
15	14	con1d 109	. . . . . . . . . . . . . . . . 17 |- ((S C_ On /\ x e. S) -> (-. x = U.S -> x e. U.S))
16		nelneq 1985	. . . . . . . . . . . . . . . . 17 |- ((x e. S /\ -. U.S e. S) -> -. x = U.S)
17	15, 16	syl5 20	. . . . . . . . . . . . . . . 16 |- ((S C_ On /\ x e. S) -> ((x e. S /\ -. U.S e. S) -> x e. U.S))
18	17	exp4b 410	. . . . . . . . . . . . . . 15 |- (S C_ On -> (x e. S -> (x e. S -> (-. U.S e. S -> x e. U.S))))
19	18	pm2.43d 79	. . . . . . . . . . . . . 14 |- (S C_ On -> (x e. S -> (-. U.S e. S -> x e. U.S)))
20	19	com23 36	. . . . . . . . . . . . 13 |- (S C_ On -> (-. U.S e. S -> (x e. S -> x e. U.S)))
21	20	imp 377	. . . . . . . . . . . 12 |- ((S C_ On /\ -. U.S e. S) -> (x e. S -> x e. U.S))
22	21	ssrdv 2622	. . . . . . . . . . 11 |- ((S C_ On /\ -. U.S e. S) -> S C_ U.S)
23	3, 22	sylan 497	. . . . . . . . . 10 |- (((S C_ On /\ -. U.S e. S) /\ S =/= (/)) -> U.S =/= (/))
24		uniss 3199	. . . . . . . . . . . . 13 |- (S C_ U.S -> U.S C_ U.U.S)
25	22, 24	syl 12	. . . . . . . . . . . 12 |- ((S C_ On /\ -. U.S e. S) -> U.S C_ U.U.S)
26		orduniss 3765	. . . . . . . . . . . . . 14 |- (Ord U.S -> U.U.S C_ U.S)
27	1, 26	syl 12	. . . . . . . . . . . . 13 |- (S C_ On -> U.U.S C_ U.S)
28	27	adantr 425	. . . . . . . . . . . 12 |- ((S C_ On /\ -. U.S e. S) -> U.U.S C_ U.S)
29	25, 28	eqssd 2633	. . . . . . . . . . 11 |- ((S C_ On /\ -. U.S e. S) -> U.S = U.U.S)
30	29	adantr 425	. . . . . . . . . 10 |- (((S C_ On /\ -. U.S e. S) /\ S =/= (/)) -> U.S = U.U.S)
31	2, 23, 30	3jca 1050	. . . . . . . . 9 |- (((S C_ On /\ -. U.S e. S) /\ S =/= (/)) -> (Ord U.S /\ U.S =/= (/) /\ U.S = U.U.S))
32		df-lim 3662	. . . . . . . . 9 |- (Lim U.S <-> (Ord U.S /\ U.S =/= (/) /\ U.S = U.U.S))
33	31, 32	sylibr 217	. . . . . . . 8 |- (((S C_ On /\ -. U.S e. S) /\ S =/= (/)) -> Lim U.S)
34	33	an1rs 547	. . . . . . 7 |- (((S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> Lim U.S)
35	34	3adantl1 1032	. . . . . 6 |- (((S e. T /\ S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> Lim U.S)
36		ssonuni 3872	. . . . . . . . . 10 |- (S e. T -> (S C_ On -> U.S e. On))
37		limeq 3669	. . . . . . . . . . . 12 |- (y = U.S -> (Lim y <-> Lim U.S))
38		fveq2 4681	. . . . . . . . . . . . 13 |- (y = U.S -> (F` y) = (F` U.S))
39		iuneq1 3269	. . . . . . . . . . . . 13 |- (y = U.S -> U_x e. y (F` x) = U_x e. U.S(F` x))
40	38, 39	eqeq12d 1899	. . . . . . . . . . . 12 |- (y = U.S -> ((F` y) = U_x e. y (F` x) <-> (F` U.S) = U_x e. U.S(F` x)))
41	37, 40	imbi12d 688	. . . . . . . . . . 11 |- (y = U.S -> ((Lim y -> (F` y) = U_x e. y (F` x)) <-> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x))))
42		onfununi.1	. . . . . . . . . . 11 |- (Lim y -> (F` y) = U_x e. y (F` x))
43	41, 42	vtoclg 2346	. . . . . . . . . 10 |- (U.S e. On -> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x)))
44	36, 43	syl6 25	. . . . . . . . 9 |- (S e. T -> (S C_ On -> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x))))
45	44	imp 377	. . . . . . . 8 |- ((S e. T /\ S C_ On) -> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x)))
46	45	3adant3 896	. . . . . . 7 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x)))
47	46	adantr 425	. . . . . 6 |- (((S e. T /\ S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> (Lim U.S -> (F` U.S) = U_x e. U.S(F` x)))
48	35, 47	mpd 29	. . . . 5 |- (((S e. T /\ S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> (F` U.S) = U_x e. U.S(F` x))
49		ssel 2615	. . . . . . . . . . . . . . . . . 18 |- (S C_ On -> (y e. S -> y e. On))
50	49	anim1d 619	. . . . . . . . . . . . . . . . 17 |- (S C_ On -> ((y e. S /\ x e. y) -> (y e. On /\ x e. y)))
51		onelon 3683	. . . . . . . . . . . . . . . . 17 |- ((y e. On /\ x e. y) -> x e. On)
52	50, 51	syl6 25	. . . . . . . . . . . . . . . 16 |- (S C_ On -> ((y e. S /\ x e. y) -> x e. On))
53	49	adantrd 427	. . . . . . . . . . . . . . . 16 |- (S C_ On -> ((y e. S /\ x e. y) -> y e. On))
54		ordelss 3674	. . . . . . . . . . . . . . . . . 18 |- ((Ord y /\ x e. y) -> x C_ y)
55	54	a1i 8	. . . . . . . . . . . . . . . . 17 |- (S C_ On -> ((Ord y /\ x e. y) -> x C_ y))
56		eloni 3667	. . . . . . . . . . . . . . . . . 18 |- (y e. On -> Ord y)
57	49, 56	syl6 25	. . . . . . . . . . . . . . . . 17 |- (S C_ On -> (y e. S -> Ord y))
58	55, 57	syland 506	. . . . . . . . . . . . . . . 16 |- (S C_ On -> ((y e. S /\ x e. y) -> x C_ y))
59	52, 53, 58	3jcad 1051	. . . . . . . . . . . . . . 15 |- (S C_ On -> ((y e. S /\ x e. y) -> (x e. On /\ y e. On /\ x C_ y)))
60		onfununi.2	. . . . . . . . . . . . . . 15 |- ((x e. On /\ y e. On /\ x C_ y) -> (F` x) C_ (F` y))
61	59, 60	syl6 25	. . . . . . . . . . . . . 14 |- (S C_ On -> ((y e. S /\ x e. y) -> (F` x) C_ (F` y)))
62	61	exp3a 405	. . . . . . . . . . . . 13 |- (S C_ On -> (y e. S -> (x e. y -> (F` x) C_ (F` y))))
63	62	reximdvai 2201	. . . . . . . . . . . 12 |- (S C_ On -> (E.y e. S x e. y -> E.y e. S (F` x) C_ (F` y)))
64		eluni2 3181	. . . . . . . . . . . 12 |- (x e. U.S <-> E.y e. S x e. y)
65	63, 64	syl5ib 223	. . . . . . . . . . 11 |- (S C_ On -> (x e. U.S -> E.y e. S (F` x) C_ (F` y)))
66		ssiun 3293	. . . . . . . . . . 11 |- (E.y e. S (F` x) C_ (F` y) -> (F` x) C_ U_y e. S (F` y))
67	65, 66	syl6 25	. . . . . . . . . 10 |- (S C_ On -> (x e. U.S -> (F` x) C_ U_y e. S (F` y)))
68	67	r19.21aiv 2175	. . . . . . . . 9 |- (S C_ On -> A.x e. U.S(F` x) C_ U_y e. S (F` y))
69		iunss 3291	. . . . . . . . 9 |- (U_x e. U.S(F` x) C_ U_y e. S (F` y) <-> A.x e. U.S(F` x) C_ U_y e. S (F` y))
70	68, 69	sylibr 217	. . . . . . . 8 |- (S C_ On -> U_x e. U.S(F` x) C_ U_y e. S (F` y))
71		fveq2 4681	. . . . . . . . 9 |- (y = x -> (F` y) = (F` x))
72	71	cbviunv 3290	. . . . . . . 8 |- U_y e. S (F` y) = U_x e. S (F` x)
73	70, 72	syl6ss 2663	. . . . . . 7 |- (S C_ On -> U_x e. U.S(F` x) C_ U_x e. S (F` x))
74	73	3ad2ant2 898	. . . . . 6 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> U_x e. U.S(F` x) C_ U_x e. S (F` x))
75	74	adantr 425	. . . . 5 |- (((S e. T /\ S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> U_x e. U.S(F` x) C_ U_x e. S (F` x))
76	48, 75	eqsstrd 2651	. . . 4 |- (((S e. T /\ S C_ On /\ S =/= (/)) /\ -. U.S e. S) -> (F` U.S) C_ U_x e. S (F` x))
77	76	ex 402	. . 3 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (-. U.S e. S -> (F` U.S) C_ U_x e. S (F` x)))
78		fveq2 4681	. . . 4 |- (x = U.S -> (F` x) = (F` U.S))
79	78	ssiun2s 3297	. . 3 |- (U.S e. S -> (F` U.S) C_ U_x e. S (F` x))
80	77, 79	pm2.61d2 143	. 2 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (F` U.S) C_ U_x e. S (F` x))
81	36	imp 377	. . . . . 6 |- ((S e. T /\ S C_ On) -> U.S e. On)
82	81	3adant3 896	. . . . 5 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> U.S e. On)
83	7	3ad2ant2 898	. . . . . 6 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (x e. S -> x e. On))
84	4	a1i 8	. . . . . 6 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (x e. S -> x C_ U.S))
85	83, 84	jcad 661	. . . . 5 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (x e. S -> (x e. On /\ x C_ U.S)))
86		sseq2 2639	. . . . . . . 8 |- (y = U.S -> (x C_ y <-> x C_ U.S))
87	86	anbi2d 678	. . . . . . 7 |- (y = U.S -> ((x e. On /\ x C_ y) <-> (x e. On /\ x C_ U.S)))
88	38	sseq2d 2645	. . . . . . 7 |- (y = U.S -> ((F` x) C_ (F` y) <-> (F` x) C_ (F` U.S)))
89	87, 88	imbi12d 688	. . . . . 6 |- (y = U.S -> (((x e. On /\ x C_ y) -> (F` x) C_ (F` y)) <-> ((x e. On /\ x C_ U.S) -> (F` x) C_ (F` U.S))))
90	60	3com12 1071	. . . . . . 7 |- ((y e. On /\ x e. On /\ x C_ y) -> (F` x) C_ (F` y))
91	90	3expib 1070	. . . . . 6 |- (y e. On -> ((x e. On /\ x C_ y) -> (F` x) C_ (F` y)))
92	89, 91	vtoclga 2352	. . . . 5 |- (U.S e. On -> ((x e. On /\ x C_ U.S) -> (F` x) C_ (F` U.S)))
93	82, 85, 92	sylsyld 32	. . . 4 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (x e. S -> (F` x) C_ (F` U.S)))
94	93	r19.21aiv 2175	. . 3 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> A.x e. S (F` x) C_ (F` U.S))
95		iunss 3291	. . 3 |- (U_x e. S (F` x) C_ (F` U.S) <-> A.x e. S (F` x) C_ (F` U.S))
96	94, 95	sylibr 217	. 2 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> U_x e. S (F` x) C_ (F` U.S))
97	80, 96	eqssd 2633	1 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (F` U.S) = U_x e. S (F` x))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  U.cuni 3177  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  ` cfv 3998

This theorem is referenced by:  onopruni 5117

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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