Theorem dyadmbl 23368

Description: Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypotheses

Ref	Expression
dyadmbl.1	|- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )
dyadmbl.2	|- G = { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) }
dyadmbl.3	|- ( ph -> A C_ ran F )

Assertion

Ref	Expression
dyadmbl	|- ( ph -> U. ( [,] " A ) e. dom vol )

Distinct variable groups:   x, y   z, w, ph   x, w, y, A, z   z, G   w, F, x, y, z

Allowed substitution hints:   ph( x, y)   G( x, y, w)

Proof of Theorem dyadmbl

Dummy variables f a b n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dyadmbl.1	. . 3 |- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )
2		dyadmbl.2	. . 3 |- G = { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) }
3		dyadmbl.3	. . 3 |- ( ph -> A C_ ran F )
4	1, 2, 3	dyadmbllem 23367	. 2 |- ( ph -> U. ( [,] " A ) = U. ( [,] " G ) )
5		isfinite 8549	. . . 4 |- ( G e. Fin <-> G ~< om )
6		iccf 12272	. . . . . 6 |- [,] : ( RR* X. RR* ) --> ~P RR*
7		ffun 6048	. . . . . 6 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> Fun [,] )
8		funiunfv 6506	. . . . . 6 |- ( Fun [,] -> U_ n e. G ( [,] ` n ) = U. ( [,] " G ) )
9	6, 7, 8	mp2b 10	. . . . 5 |- U_ n e. G ( [,] ` n ) = U. ( [,] " G )
10		simpr 477	. . . . . 6 |- ( ( ph /\ G e. Fin ) -> G e. Fin )
11		ssrab2 3687	. . . . . . . . . . . . . . . 16 |- { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) } C_ A
12	2, 11	eqsstri 3635	. . . . . . . . . . . . . . 15 |- G C_ A
13	12, 3	syl5ss 3614	. . . . . . . . . . . . . 14 |- ( ph -> G C_ ran F )
14	1	dyadf 23359	. . . . . . . . . . . . . . . 16 |- F : ( ZZ X. NN0 ) --> ( <_ i^i ( RR X. RR ) )
15		frn 6053	. . . . . . . . . . . . . . . 16 |- ( F : ( ZZ X. NN0 ) --> ( <_ i^i ( RR X. RR ) ) -> ran F C_ ( <_ i^i ( RR X. RR ) ) )
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . 15 |- ran F C_ ( <_ i^i ( RR X. RR ) )
17		inss2 3834	. . . . . . . . . . . . . . 15 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
18	16, 17	sstri 3612	. . . . . . . . . . . . . 14 |- ran F C_ ( RR X. RR )
19	13, 18	syl6ss 3615	. . . . . . . . . . . . 13 |- ( ph -> G C_ ( RR X. RR ) )
20	19	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ G e. Fin ) -> G C_ ( RR X. RR ) )
21	20	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> n e. ( RR X. RR ) )
22		1st2nd2 7205	. . . . . . . . . . 11 |- ( n e. ( RR X. RR ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
23	21, 22	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> n = <. ( 1st ` n ) , ( 2nd ` n ) >. )
24	23	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( [,] ` n ) = ( [,] ` <. ( 1st ` n ) , ( 2nd ` n ) >. ) )
25		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` n ) [,] ( 2nd ` n ) ) = ( [,] ` <. ( 1st ` n ) , ( 2nd ` n ) >. )
26	24, 25	syl6eqr 2674	. . . . . . . 8 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( [,] ` n ) = ( ( 1st ` n ) [,] ( 2nd ` n ) ) )
27		xp1st 7198	. . . . . . . . . 10 |- ( n e. ( RR X. RR ) -> ( 1st ` n ) e. RR )
28	21, 27	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( 1st ` n ) e. RR )
29		xp2nd 7199	. . . . . . . . . 10 |- ( n e. ( RR X. RR ) -> ( 2nd ` n ) e. RR )
30	21, 29	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( 2nd ` n ) e. RR )
31		iccmbl 23334	. . . . . . . . 9 |- ( ( ( 1st ` n ) e. RR /\ ( 2nd ` n ) e. RR ) -> ( ( 1st ` n ) [,] ( 2nd ` n ) ) e. dom vol )
32	28, 30, 31	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( ( 1st ` n ) [,] ( 2nd ` n ) ) e. dom vol )
33	26, 32	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ G e. Fin ) /\ n e. G ) -> ( [,] ` n ) e. dom vol )
34	33	ralrimiva 2966	. . . . . 6 |- ( ( ph /\ G e. Fin ) -> A. n e. G ( [,] ` n ) e. dom vol )
35		finiunmbl 23312	. . . . . 6 |- ( ( G e. Fin /\ A. n e. G ( [,] ` n ) e. dom vol ) -> U_ n e. G ( [,] ` n ) e. dom vol )
36	10, 34, 35	syl2anc 693	. . . . 5 |- ( ( ph /\ G e. Fin ) -> U_ n e. G ( [,] ` n ) e. dom vol )
37	9, 36	syl5eqelr 2706	. . . 4 |- ( ( ph /\ G e. Fin ) -> U. ( [,] " G ) e. dom vol )
38	5, 37	sylan2br 493	. . 3 |- ( ( ph /\ G ~< om ) -> U. ( [,] " G ) e. dom vol )
39		nnenom 12779	. . . . . . 7 |- NN ~~ om
40		ensym 8005	. . . . . . 7 |- ( G ~~ om -> om ~~ G )
41		entr 8008	. . . . . . 7 |- ( ( NN ~~ om /\ om ~~ G ) -> NN ~~ G )
42	39, 40, 41	sylancr 695	. . . . . 6 |- ( G ~~ om -> NN ~~ G )
43		bren 7964	. . . . . 6 |- ( NN ~~ G <-> E. f f : NN -1-1-onto-> G )
44	42, 43	sylib 208	. . . . 5 |- ( G ~~ om -> E. f f : NN -1-1-onto-> G )
45		rnco2 5642	. . . . . . . . . 10 |- ran ( [,] o. f ) = ( [,] " ran f )
46		f1ofo 6144	. . . . . . . . . . . . 13 |- ( f : NN -1-1-onto-> G -> f : NN -onto-> G )
47	46	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> f : NN -onto-> G )
48		forn 6118	. . . . . . . . . . . 12 |- ( f : NN -onto-> G -> ran f = G )
49	47, 48	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> ran f = G )
50	49	imaeq2d 5466	. . . . . . . . . 10 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> ( [,] " ran f ) = ( [,] " G ) )
51	45, 50	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> ran ( [,] o. f ) = ( [,] " G ) )
52	51	unieqd 4446	. . . . . . . 8 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> U. ran ( [,] o. f ) = U. ( [,] " G ) )
53		f1of 6137	. . . . . . . . . 10 |- ( f : NN -1-1-onto-> G -> f : NN --> G )
54	13, 16	syl6ss 3615	. . . . . . . . . 10 |- ( ph -> G C_ ( <_ i^i ( RR X. RR ) ) )
55		fss 6056	. . . . . . . . . 10 |- ( ( f : NN --> G /\ G C_ ( <_ i^i ( RR X. RR ) ) ) -> f : NN --> ( <_ i^i ( RR X. RR ) ) )
56	53, 54, 55	syl2anr 495	. . . . . . . . 9 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> f : NN --> ( <_ i^i ( RR X. RR ) ) )
57		fss 6056	. . . . . . . . . . . . . . 15 |- ( ( f : NN --> G /\ G C_ ran F ) -> f : NN --> ran F )
58	53, 13, 57	syl2anr 495	. . . . . . . . . . . . . 14 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> f : NN --> ran F )
59		simpl 473	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ b e. NN ) -> a e. NN )
60		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran F /\ a e. NN ) -> ( f ` a ) e. ran F )
61	58, 59, 60	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` a ) e. ran F )
62		simpr 477	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ b e. NN ) -> b e. NN )
63		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran F /\ b e. NN ) -> ( f ` b ) e. ran F )
64	58, 62, 63	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` b ) e. ran F )
65	1	dyaddisj 23364	. . . . . . . . . . . . 13 |- ( ( ( f ` a ) e. ran F /\ ( f ` b ) e. ran F ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) \/ ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
66	61, 64, 65	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) \/ ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
67	53	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> f : NN --> G )
68		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 |- ( ( f : NN --> G /\ b e. NN ) -> ( f ` b ) e. G )
69	67, 62, 68	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` b ) e. G )
70	12, 69	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` b ) e. A )
71		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 |- ( ( f : NN --> G /\ a e. NN ) -> ( f ` a ) e. G )
72	67, 59, 71	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` a ) e. G )
73		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( z = ( f ` a ) -> ( [,] ` z ) = ( [,] ` ( f ` a ) ) )
74	73	sseq1d 3632	. . . . . . . . . . . . . . . . . . . 20 |- ( z = ( f ` a ) -> ( ( [,] ` z ) C_ ( [,] ` w ) <-> ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) ) )
75		eqeq1 2626	. . . . . . . . . . . . . . . . . . . 20 |- ( z = ( f ` a ) -> ( z = w <-> ( f ` a ) = w ) )
76	74, 75	imbi12d 334	. . . . . . . . . . . . . . . . . . 19 |- ( z = ( f ` a ) -> ( ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) ) )
77	76	ralbidv 2986	. . . . . . . . . . . . . . . . . 18 |- ( z = ( f ` a ) -> ( A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> A. w e. A ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) ) )
78	77, 2	elrab2 3366	. . . . . . . . . . . . . . . . 17 |- ( ( f ` a ) e. G <-> ( ( f ` a ) e. A /\ A. w e. A ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) ) )
79	78	simprbi 480	. . . . . . . . . . . . . . . 16 |- ( ( f ` a ) e. G -> A. w e. A ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) )
80	72, 79	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> A. w e. A ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) )
81		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( w = ( f ` b ) -> ( [,] ` w ) = ( [,] ` ( f ` b ) ) )
82	81	sseq2d 3633	. . . . . . . . . . . . . . . . 17 |- ( w = ( f ` b ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) <-> ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) ) )
83		eqeq2 2633	. . . . . . . . . . . . . . . . 17 |- ( w = ( f ` b ) -> ( ( f ` a ) = w <-> ( f ` a ) = ( f ` b ) ) )
84	82, 83	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( w = ( f ` b ) -> ( ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) <-> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) -> ( f ` a ) = ( f ` b ) ) ) )
85	84	rspcv 3305	. . . . . . . . . . . . . . 15 |- ( ( f ` b ) e. A -> ( A. w e. A ( ( [,] ` ( f ` a ) ) C_ ( [,] ` w ) -> ( f ` a ) = w ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) -> ( f ` a ) = ( f ` b ) ) ) )
86	70, 80, 85	sylc 65	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) -> ( f ` a ) = ( f ` b ) ) )
87		f1of1 6136	. . . . . . . . . . . . . . . . 17 |- ( f : NN -1-1-onto-> G -> f : NN -1-1-> G )
88	87	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> f : NN -1-1-> G )
89		f1fveq 6519	. . . . . . . . . . . . . . . 16 |- ( ( f : NN -1-1-> G /\ ( a e. NN /\ b e. NN ) ) -> ( ( f ` a ) = ( f ` b ) <-> a = b ) )
90	88, 89	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( f ` a ) = ( f ` b ) <-> a = b ) )
91		orc 400	. . . . . . . . . . . . . . 15 |- ( a = b -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
92	90, 91	syl6bi 243	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( f ` a ) = ( f ` b ) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) ) )
93	86, 92	syld 47	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) ) )
94	12, 72	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( f ` a ) e. A )
95		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( z = ( f ` b ) -> ( [,] ` z ) = ( [,] ` ( f ` b ) ) )
96	95	sseq1d 3632	. . . . . . . . . . . . . . . . . . . 20 |- ( z = ( f ` b ) -> ( ( [,] ` z ) C_ ( [,] ` w ) <-> ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) ) )
97		eqeq1 2626	. . . . . . . . . . . . . . . . . . . 20 |- ( z = ( f ` b ) -> ( z = w <-> ( f ` b ) = w ) )
98	96, 97	imbi12d 334	. . . . . . . . . . . . . . . . . . 19 |- ( z = ( f ` b ) -> ( ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) ) )
99	98	ralbidv 2986	. . . . . . . . . . . . . . . . . 18 |- ( z = ( f ` b ) -> ( A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> A. w e. A ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) ) )
100	99, 2	elrab2 3366	. . . . . . . . . . . . . . . . 17 |- ( ( f ` b ) e. G <-> ( ( f ` b ) e. A /\ A. w e. A ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) ) )
101	100	simprbi 480	. . . . . . . . . . . . . . . 16 |- ( ( f ` b ) e. G -> A. w e. A ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) )
102	69, 101	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> A. w e. A ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) )
103		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( w = ( f ` a ) -> ( [,] ` w ) = ( [,] ` ( f ` a ) ) )
104	103	sseq2d 3633	. . . . . . . . . . . . . . . . 17 |- ( w = ( f ` a ) -> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) <-> ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) ) )
105		eqeq2 2633	. . . . . . . . . . . . . . . . . 18 |- ( w = ( f ` a ) -> ( ( f ` b ) = w <-> ( f ` b ) = ( f ` a ) ) )
106		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( ( f ` b ) = ( f ` a ) <-> ( f ` a ) = ( f ` b ) )
107	105, 106	syl6bb 276	. . . . . . . . . . . . . . . . 17 |- ( w = ( f ` a ) -> ( ( f ` b ) = w <-> ( f ` a ) = ( f ` b ) ) )
108	104, 107	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( w = ( f ` a ) -> ( ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) <-> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) -> ( f ` a ) = ( f ` b ) ) ) )
109	108	rspcv 3305	. . . . . . . . . . . . . . 15 |- ( ( f ` a ) e. A -> ( A. w e. A ( ( [,] ` ( f ` b ) ) C_ ( [,] ` w ) -> ( f ` b ) = w ) -> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) -> ( f ` a ) = ( f ` b ) ) ) )
110	94, 102, 109	sylc 65	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) -> ( f ` a ) = ( f ` b ) ) )
111	110, 92	syld 47	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) ) )
112		olc 399	. . . . . . . . . . . . . 14 |- ( ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
113	112	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) ) )
114	93, 111, 113	3jaod 1392	. . . . . . . . . . . 12 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( ( ( [,] ` ( f ` a ) ) C_ ( [,] ` ( f ` b ) ) \/ ( [,] ` ( f ` b ) ) C_ ( [,] ` ( f ` a ) ) \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) ) )
115	66, 114	mpd 15	. . . . . . . . . . 11 |- ( ( ( ph /\ f : NN -1-1-onto-> G ) /\ ( a e. NN /\ b e. NN ) ) -> ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
116	115	ralrimivva 2971	. . . . . . . . . 10 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> A. a e. NN A. b e. NN ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
117		fveq2 6191	. . . . . . . . . . . 12 |- ( a = b -> ( f ` a ) = ( f ` b ) )
118	117	fveq2d 6195	. . . . . . . . . . 11 |- ( a = b -> ( (,) ` ( f ` a ) ) = ( (,) ` ( f ` b ) ) )
119	118	disjor 4634	. . . . . . . . . 10 |- (Disj a e. NN ( (,) ` ( f ` a ) ) <-> A. a e. NN A. b e. NN ( a = b \/ ( ( (,) ` ( f ` a ) ) i^i ( (,) ` ( f ` b ) ) ) = (/) ) )
120	116, 119	sylibr 224	. . . . . . . . 9 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> Disj a e. NN ( (,) ` ( f ` a ) ) )
121		eqid 2622	. . . . . . . . 9 |- seq 1 ( + , ( ( abs o. - ) o. f ) ) = seq 1 ( + , ( ( abs o. - ) o. f ) )
122	56, 120, 121	uniiccmbl 23358	. . . . . . . 8 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> U. ran ( [,] o. f ) e. dom vol )
123	52, 122	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ f : NN -1-1-onto-> G ) -> U. ( [,] " G ) e. dom vol )
124	123	ex 450	. . . . . 6 |- ( ph -> ( f : NN -1-1-onto-> G -> U. ( [,] " G ) e. dom vol ) )
125	124	exlimdv 1861	. . . . 5 |- ( ph -> ( E. f f : NN -1-1-onto-> G -> U. ( [,] " G ) e. dom vol ) )
126	44, 125	syl5 34	. . . 4 |- ( ph -> ( G ~~ om -> U. ( [,] " G ) e. dom vol ) )
127	126	imp 445	. . 3 |- ( ( ph /\ G ~~ om ) -> U. ( [,] " G ) e. dom vol )
128		reex 10027	. . . . . . . . 9 |- RR e. _V
129	128, 128	xpex 6962	. . . . . . . 8 |- ( RR X. RR ) e. _V
130	129	inex2 4800	. . . . . . 7 |- ( <_ i^i ( RR X. RR ) ) e. _V
131	130, 16	ssexi 4803	. . . . . 6 |- ran F e. _V
132		ssdomg 8001	. . . . . 6 |- ( ran F e. _V -> ( G C_ ran F -> G ~<_ ran F ) )
133	131, 13, 132	mpsyl 68	. . . . 5 |- ( ph -> G ~<_ ran F )
134		omelon 8543	. . . . . . . 8 |- om e. On
135		znnen 14941	. . . . . . . . . . . 12 |- ZZ ~~ NN
136	135, 39	entri 8010	. . . . . . . . . . 11 |- ZZ ~~ om
137		nn0ennn 12778	. . . . . . . . . . . 12 |- NN0 ~~ NN
138	137, 39	entri 8010	. . . . . . . . . . 11 |- NN0 ~~ om
139		xpen 8123	. . . . . . . . . . 11 |- ( ( ZZ ~~ om /\ NN0 ~~ om ) -> ( ZZ X. NN0 ) ~~ ( om X. om ) )
140	136, 138, 139	mp2an 708	. . . . . . . . . 10 |- ( ZZ X. NN0 ) ~~ ( om X. om )
141		xpomen 8838	. . . . . . . . . 10 |- ( om X. om ) ~~ om
142	140, 141	entri 8010	. . . . . . . . 9 |- ( ZZ X. NN0 ) ~~ om
143	142	ensymi 8006	. . . . . . . 8 |- om ~~ ( ZZ X. NN0 )
144		isnumi 8772	. . . . . . . 8 |- ( ( om e. On /\ om ~~ ( ZZ X. NN0 ) ) -> ( ZZ X. NN0 ) e. dom card )
145	134, 143, 144	mp2an 708	. . . . . . 7 |- ( ZZ X. NN0 ) e. dom card
146		ffn 6045	. . . . . . . . 9 |- ( F : ( ZZ X. NN0 ) --> ( <_ i^i ( RR X. RR ) ) -> F Fn ( ZZ X. NN0 ) )
147	14, 146	ax-mp 5	. . . . . . . 8 |- F Fn ( ZZ X. NN0 )
148		dffn4 6121	. . . . . . . 8 |- ( F Fn ( ZZ X. NN0 ) <-> F : ( ZZ X. NN0 ) -onto-> ran F )
149	147, 148	mpbi 220	. . . . . . 7 |- F : ( ZZ X. NN0 ) -onto-> ran F
150		fodomnum 8880	. . . . . . 7 |- ( ( ZZ X. NN0 ) e. dom card -> ( F : ( ZZ X. NN0 ) -onto-> ran F -> ran F ~<_ ( ZZ X. NN0 ) ) )
151	145, 149, 150	mp2 9	. . . . . 6 |- ran F ~<_ ( ZZ X. NN0 )
152		domentr 8015	. . . . . 6 |- ( ( ran F ~<_ ( ZZ X. NN0 ) /\ ( ZZ X. NN0 ) ~~ om ) -> ran F ~<_ om )
153	151, 142, 152	mp2an 708	. . . . 5 |- ran F ~<_ om
154		domtr 8009	. . . . 5 |- ( ( G ~<_ ran F /\ ran F ~<_ om ) -> G ~<_ om )
155	133, 153, 154	sylancl 694	. . . 4 |- ( ph -> G ~<_ om )
156		brdom2 7985	. . . 4 |- ( G ~<_ om <-> ( G ~< om \/ G ~~ om ) )
157	155, 156	sylib 208	. . 3 |- ( ph -> ( G ~< om \/ G ~~ om ) )
158	38, 127, 157	mpjaodan 827	. 2 |- ( ph -> U. ( [,] " G ) e. dom vol )
159	4, 158	eqeltrd 2701	1 |- ( ph -> U. ( [,] " A ) e. dom vol )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   <.cop 4183   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   2ndc2nd 7167   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   (,)cioo 12175   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   volcvol 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  opnmbllem  23369