Theorem hartogslem1 8447

Description: Lemma for hartogs 8449. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
hartogslem.2	|- F = { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
hartogslem.3	|- R = { <. s , t >. | E. w e. y E. z e. y ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) }

Assertion

Ref	Expression
hartogslem1	|- ( dom F C_ ~P ( A X. A ) /\ Fun F /\ ( A e. V -> ran F = { x e. On | x ~<_ A } ) )

Distinct variable groups:   f, s, t, w, y, z   f, r, x, A, y   R, r, x   V, r, y

Allowed substitution hints:   A( z, w, t, s)   R( y, z, w, t, f, s)   F( x, y, z, w, t, f, s, r)   V( x, z, w, t, f, s)

Proof of Theorem hartogslem1

Step	Hyp	Ref	Expression
1		hartogslem.2	. . . . 5 |- F = { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
2	1	dmeqi 5325	. . . 4 |- dom F = dom { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
3		dmopab 5335	. . . 4 |- dom { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = { r | E. y ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
4	2, 3	eqtri 2644	. . 3 |- dom F = { r | E. y ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
5		simp3 1063	. . . . . . . 8 |- ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) -> r C_ ( dom r X. dom r ) )
6		simp1 1061	. . . . . . . . 9 |- ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) -> dom r C_ A )
7		xpss12 5225	. . . . . . . . 9 |- ( ( dom r C_ A /\ dom r C_ A ) -> ( dom r X. dom r ) C_ ( A X. A ) )
8	6, 6, 7	syl2anc 693	. . . . . . . 8 |- ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) -> ( dom r X. dom r ) C_ ( A X. A ) )
9	5, 8	sstrd 3613	. . . . . . 7 |- ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) -> r C_ ( A X. A ) )
10		selpw 4165	. . . . . . 7 |- ( r e. ~P ( A X. A ) <-> r C_ ( A X. A ) )
11	9, 10	sylibr 224	. . . . . 6 |- ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) -> r e. ~P ( A X. A ) )
12	11	ad2antrr 762	. . . . 5 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> r e. ~P ( A X. A ) )
13	12	exlimiv 1858	. . . 4 |- ( E. y ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> r e. ~P ( A X. A ) )
14	13	abssi 3677	. . 3 |- { r | E. y ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } C_ ~P ( A X. A )
15	4, 14	eqsstri 3635	. 2 |- dom F C_ ~P ( A X. A )
16		funopab4 5925	. . 3 |- Fun { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
17	1	funeqi 5909	. . 3 |- ( Fun F <-> Fun { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
18	16, 17	mpbir 221	. 2 |- Fun F
19		breq1 4656	. . . . . 6 |- ( x = y -> ( x ~<_ A <-> y ~<_ A ) )
20	19	elrab 3363	. . . . 5 |- ( y e. { x e. On | x ~<_ A } <-> ( y e. On /\ y ~<_ A ) )
21		brdomi 7966	. . . . . . 7 |- ( y ~<_ A -> E. f f : y -1-1-> A )
22		f1f 6101	. . . . . . . . . . . . . 14 |- ( f : y -1-1-> A -> f : y --> A )
23	22	adantl 482	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> f : y --> A )
24		frn 6053	. . . . . . . . . . . . 13 |- ( f : y --> A -> ran f C_ A )
25	23, 24	syl 17	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ran f C_ A )
26		resss 5422	. . . . . . . . . . . . . . 15 |- ( _I |` ran f ) C_ _I
27		ssun2 3777	. . . . . . . . . . . . . . 15 |- _I C_ ( R u. _I )
28	26, 27	sstri 3612	. . . . . . . . . . . . . 14 |- ( _I |` ran f ) C_ ( R u. _I )
29		f1oi 6174	. . . . . . . . . . . . . . 15 |- ( _I |` ran f ) : ran f -1-1-onto-> ran f
30		f1of 6137	. . . . . . . . . . . . . . 15 |- ( ( _I |` ran f ) : ran f -1-1-onto-> ran f -> ( _I |` ran f ) : ran f --> ran f )
31		fssxp 6060	. . . . . . . . . . . . . . 15 |- ( ( _I |` ran f ) : ran f --> ran f -> ( _I |` ran f ) C_ ( ran f X. ran f ) )
32	29, 30, 31	mp2b 10	. . . . . . . . . . . . . 14 |- ( _I |` ran f ) C_ ( ran f X. ran f )
33	28, 32	ssini 3836	. . . . . . . . . . . . 13 |- ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) )
34	33	a1i 11	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) )
35		inss2 3834	. . . . . . . . . . . . 13 |- ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f )
36	35	a1i 11	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) )
37	25, 34, 36	3jca 1242	. . . . . . . . . . 11 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ran f C_ A /\ ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) /\ ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) )
38		eloni 5733	. . . . . . . . . . . . . . 15 |- ( y e. On -> Ord y )
39		ordwe 5736	. . . . . . . . . . . . . . 15 |- ( Ord y -> _E We y )
40	38, 39	syl 17	. . . . . . . . . . . . . 14 |- ( y e. On -> _E We y )
41	40	adantr 481	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> _E We y )
42		f1f1orn 6148	. . . . . . . . . . . . . . . . 17 |- ( f : y -1-1-> A -> f : y -1-1-onto-> ran f )
43	42	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( y e. On /\ f : y -1-1-> A ) -> f : y -1-1-onto-> ran f )
44		hartogslem.3	. . . . . . . . . . . . . . . 16 |- R = { <. s , t >. | E. w e. y E. z e. y ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) }
45		f1oiso 6601	. . . . . . . . . . . . . . . 16 |- ( ( f : y -1-1-onto-> ran f /\ R = { <. s , t >. | E. w e. y E. z e. y ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) } ) -> f Isom _E , R ( y , ran f ) )
46	43, 44, 45	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ( y e. On /\ f : y -1-1-> A ) -> f Isom _E , R ( y , ran f ) )
47		isores2 6583	. . . . . . . . . . . . . . 15 |- ( f Isom _E , R ( y , ran f ) <-> f Isom _E , ( R i^i ( ran f X. ran f ) ) ( y , ran f ) )
48	46, 47	sylib 208	. . . . . . . . . . . . . 14 |- ( ( y e. On /\ f : y -1-1-> A ) -> f Isom _E , ( R i^i ( ran f X. ran f ) ) ( y , ran f ) )
49		isowe 6599	. . . . . . . . . . . . . 14 |- ( f Isom _E , ( R i^i ( ran f X. ran f ) ) ( y , ran f ) -> ( _E We y <-> ( R i^i ( ran f X. ran f ) ) We ran f ) )
50	48, 49	syl 17	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( _E We y <-> ( R i^i ( ran f X. ran f ) ) We ran f ) )
51	41, 50	mpbid 222	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( R i^i ( ran f X. ran f ) ) We ran f )
52		weso 5105	. . . . . . . . . . . . . . . . . . 19 |- ( ( R i^i ( ran f X. ran f ) ) We ran f -> ( R i^i ( ran f X. ran f ) ) Or ran f )
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( R i^i ( ran f X. ran f ) ) Or ran f )
54		inss2 3834	. . . . . . . . . . . . . . . . . . . 20 |- ( R i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f )
55	54	brel 5168	. . . . . . . . . . . . . . . . . . 19 |- ( x ( R i^i ( ran f X. ran f ) ) x -> ( x e. ran f /\ x e. ran f ) )
56	55	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( x ( R i^i ( ran f X. ran f ) ) x -> x e. ran f )
57		sonr 5056	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R i^i ( ran f X. ran f ) ) Or ran f /\ x e. ran f ) -> -. x ( R i^i ( ran f X. ran f ) ) x )
58	53, 56, 57	syl2an 494	. . . . . . . . . . . . . . . . 17 |- ( ( ( y e. On /\ f : y -1-1-> A ) /\ x ( R i^i ( ran f X. ran f ) ) x ) -> -. x ( R i^i ( ran f X. ran f ) ) x )
59	58	pm2.01da 458	. . . . . . . . . . . . . . . 16 |- ( ( y e. On /\ f : y -1-1-> A ) -> -. x ( R i^i ( ran f X. ran f ) ) x )
60	59	alrimiv 1855	. . . . . . . . . . . . . . 15 |- ( ( y e. On /\ f : y -1-1-> A ) -> A. x -. x ( R i^i ( ran f X. ran f ) ) x )
61		intirr 5514	. . . . . . . . . . . . . . 15 |- ( ( ( R i^i ( ran f X. ran f ) ) i^i _I ) = (/) <-> A. x -. x ( R i^i ( ran f X. ran f ) ) x )
62	60, 61	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ( R i^i ( ran f X. ran f ) ) i^i _I ) = (/) )
63		disj3 4021	. . . . . . . . . . . . . 14 |- ( ( ( R i^i ( ran f X. ran f ) ) i^i _I ) = (/) <-> ( R i^i ( ran f X. ran f ) ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) )
64	62, 63	sylib 208	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( R i^i ( ran f X. ran f ) ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) )
65		weeq1 5102	. . . . . . . . . . . . 13 |- ( ( R i^i ( ran f X. ran f ) ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) -> ( ( R i^i ( ran f X. ran f ) ) We ran f <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) )
66	64, 65	syl 17	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ( R i^i ( ran f X. ran f ) ) We ran f <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) )
67	51, 66	mpbid 222	. . . . . . . . . . 11 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f )
68	38	adantr 481	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> Ord y )
69		isoeq3 6569	. . . . . . . . . . . . . . 15 |- ( ( R i^i ( ran f X. ran f ) ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) -> ( f Isom _E , ( R i^i ( ran f X. ran f ) ) ( y , ran f ) <-> f Isom _E , ( ( R i^i ( ran f X. ran f ) ) \ _I ) ( y , ran f ) ) )
70	64, 69	syl 17	. . . . . . . . . . . . . 14 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( f Isom _E , ( R i^i ( ran f X. ran f ) ) ( y , ran f ) <-> f Isom _E , ( ( R i^i ( ran f X. ran f ) ) \ _I ) ( y , ran f ) ) )
71	48, 70	mpbid 222	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> f Isom _E , ( ( R i^i ( ran f X. ran f ) ) \ _I ) ( y , ran f ) )
72		vex 3203	. . . . . . . . . . . . . . . 16 |- f e. _V
73	72	rnex 7100	. . . . . . . . . . . . . . 15 |- ran f e. _V
74		exse 5078	. . . . . . . . . . . . . . 15 |- ( ran f e. _V -> ( ( R i^i ( ran f X. ran f ) ) \ _I ) Se ran f )
75	73, 74	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( R i^i ( ran f X. ran f ) ) \ _I ) Se ran f
76		eqid 2622	. . . . . . . . . . . . . . 15 |- OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f )
77	76	oieu 8444	. . . . . . . . . . . . . 14 |- ( ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f /\ ( ( R i^i ( ran f X. ran f ) ) \ _I ) Se ran f ) -> ( ( Ord y /\ f Isom _E , ( ( R i^i ( ran f X. ran f ) ) \ _I ) ( y , ran f ) ) <-> ( y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) /\ f = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) ) )
78	67, 75, 77	sylancl 694	. . . . . . . . . . . . 13 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( ( Ord y /\ f Isom _E , ( ( R i^i ( ran f X. ran f ) ) \ _I ) ( y , ran f ) ) <-> ( y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) /\ f = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) ) )
79	68, 71, 78	mpbi2and 956	. . . . . . . . . . . 12 |- ( ( y e. On /\ f : y -1-1-> A ) -> ( y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) /\ f = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) )
80	79	simpld 475	. . . . . . . . . . 11 |- ( ( y e. On /\ f : y -1-1-> A ) -> y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) )
81	73, 73	xpex 6962	. . . . . . . . . . . . 13 |- ( ran f X. ran f ) e. _V
82	81	inex2 4800	. . . . . . . . . . . 12 |- ( ( R u. _I ) i^i ( ran f X. ran f ) ) e. _V
83		sseq1 3626	. . . . . . . . . . . . . . . . . . . 20 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( r C_ ( ran f X. ran f ) <-> ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) )
84	35, 83	mpbiri 248	. . . . . . . . . . . . . . . . . . 19 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> r C_ ( ran f X. ran f ) )
85		dmss 5323	. . . . . . . . . . . . . . . . . . 19 |- ( r C_ ( ran f X. ran f ) -> dom r C_ dom ( ran f X. ran f ) )
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> dom r C_ dom ( ran f X. ran f ) )
87		dmxpid 5345	. . . . . . . . . . . . . . . . . 18 |- dom ( ran f X. ran f ) = ran f
88	86, 87	syl6sseq 3651	. . . . . . . . . . . . . . . . 17 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> dom r C_ ran f )
89		dmresi 5457	. . . . . . . . . . . . . . . . . 18 |- dom ( _I |` ran f ) = ran f
90		sseq2 3627	. . . . . . . . . . . . . . . . . . . 20 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( _I |` ran f ) C_ r <-> ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) ) )
91	33, 90	mpbiri 248	. . . . . . . . . . . . . . . . . . 19 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( _I |` ran f ) C_ r )
92		dmss 5323	. . . . . . . . . . . . . . . . . . 19 |- ( ( _I |` ran f ) C_ r -> dom ( _I |` ran f ) C_ dom r )
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> dom ( _I |` ran f ) C_ dom r )
94	89, 93	syl5eqssr 3650	. . . . . . . . . . . . . . . . 17 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ran f C_ dom r )
95	88, 94	eqssd 3620	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> dom r = ran f )
96	95	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( dom r C_ A <-> ran f C_ A ) )
97	95	reseq2d 5396	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( _I |` dom r ) = ( _I |` ran f ) )
98		id 22	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) )
99	97, 98	sseq12d 3634	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( _I |` dom r ) C_ r <-> ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) ) )
100	95	sqxpeqd 5141	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( dom r X. dom r ) = ( ran f X. ran f ) )
101	98, 100	sseq12d 3634	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( r C_ ( dom r X. dom r ) <-> ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) )
102	96, 99, 101	3anbi123d 1399	. . . . . . . . . . . . . 14 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) <-> ( ran f C_ A /\ ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) /\ ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) ) )
103		difeq1 3721	. . . . . . . . . . . . . . . . 17 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( r \ _I ) = ( ( ( R u. _I ) i^i ( ran f X. ran f ) ) \ _I ) )
104		difun2 4048	. . . . . . . . . . . . . . . . . . 19 |- ( ( R u. _I ) \ _I ) = ( R \ _I )
105	104	ineq1i 3810	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R u. _I ) \ _I ) i^i ( ran f X. ran f ) ) = ( ( R \ _I ) i^i ( ran f X. ran f ) )
106		indif1 3871	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R u. _I ) \ _I ) i^i ( ran f X. ran f ) ) = ( ( ( R u. _I ) i^i ( ran f X. ran f ) ) \ _I )
107		indif1 3871	. . . . . . . . . . . . . . . . . 18 |- ( ( R \ _I ) i^i ( ran f X. ran f ) ) = ( ( R i^i ( ran f X. ran f ) ) \ _I )
108	105, 106, 107	3eqtr3i 2652	. . . . . . . . . . . . . . . . 17 |- ( ( ( R u. _I ) i^i ( ran f X. ran f ) ) \ _I ) = ( ( R i^i ( ran f X. ran f ) ) \ _I )
109	103, 108	syl6eq 2672	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( r \ _I ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) )
110		weeq1 5102	. . . . . . . . . . . . . . . 16 |- ( ( r \ _I ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) -> ( ( r \ _I ) We dom r <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We dom r ) )
111	109, 110	syl 17	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( r \ _I ) We dom r <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We dom r ) )
112		weeq2 5103	. . . . . . . . . . . . . . . 16 |- ( dom r = ran f -> ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) We dom r <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) )
113	95, 112	syl 17	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) We dom r <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) )
114	111, 113	bitrd 268	. . . . . . . . . . . . . 14 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( r \ _I ) We dom r <-> ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) )
115	102, 114	anbi12d 747	. . . . . . . . . . . . 13 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) <-> ( ( ran f C_ A /\ ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) /\ ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) /\ ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) ) )
116		oieq1 8417	. . . . . . . . . . . . . . . . 17 |- ( ( r \ _I ) = ( ( R i^i ( ran f X. ran f ) ) \ _I ) -> OrdIso ( ( r \ _I ) , dom r ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , dom r ) )
117	109, 116	syl 17	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> OrdIso ( ( r \ _I ) , dom r ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , dom r ) )
118		oieq2 8418	. . . . . . . . . . . . . . . . 17 |- ( dom r = ran f -> OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , dom r ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) )
119	95, 118	syl 17	. . . . . . . . . . . . . . . 16 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , dom r ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) )
120	117, 119	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> OrdIso ( ( r \ _I ) , dom r ) = OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) )
121	120	dmeqd 5326	. . . . . . . . . . . . . 14 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> dom OrdIso ( ( r \ _I ) , dom r ) = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) )
122	121	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( y = dom OrdIso ( ( r \ _I ) , dom r ) <-> y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) )
123	115, 122	anbi12d 747	. . . . . . . . . . . 12 |- ( r = ( ( R u. _I ) i^i ( ran f X. ran f ) ) -> ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) <-> ( ( ( ran f C_ A /\ ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) /\ ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) /\ ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) /\ y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) ) )
124	82, 123	spcev 3300	. . . . . . . . . . 11 |- ( ( ( ( ran f C_ A /\ ( _I |` ran f ) C_ ( ( R u. _I ) i^i ( ran f X. ran f ) ) /\ ( ( R u. _I ) i^i ( ran f X. ran f ) ) C_ ( ran f X. ran f ) ) /\ ( ( R i^i ( ran f X. ran f ) ) \ _I ) We ran f ) /\ y = dom OrdIso ( ( ( R i^i ( ran f X. ran f ) ) \ _I ) , ran f ) ) -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) )
125	37, 67, 80, 124	syl21anc 1325	. . . . . . . . . 10 |- ( ( y e. On /\ f : y -1-1-> A ) -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) )
126	125	ex 450	. . . . . . . . 9 |- ( y e. On -> ( f : y -1-1-> A -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) )
127	126	exlimdv 1861	. . . . . . . 8 |- ( y e. On -> ( E. f f : y -1-1-> A -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) )
128	127	imp 445	. . . . . . 7 |- ( ( y e. On /\ E. f f : y -1-1-> A ) -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) )
129	21, 128	sylan2 491	. . . . . 6 |- ( ( y e. On /\ y ~<_ A ) -> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) )
130		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> y = dom OrdIso ( ( r \ _I ) , dom r ) )
131		vex 3203	. . . . . . . . . . . . 13 |- r e. _V
132	131	dmex 7099	. . . . . . . . . . . 12 |- dom r e. _V
133		eqid 2622	. . . . . . . . . . . . 13 |- OrdIso ( ( r \ _I ) , dom r ) = OrdIso ( ( r \ _I ) , dom r )
134	133	oion 8441	. . . . . . . . . . . 12 |- ( dom r e. _V -> dom OrdIso ( ( r \ _I ) , dom r ) e. On )
135	132, 134	ax-mp 5	. . . . . . . . . . 11 |- dom OrdIso ( ( r \ _I ) , dom r ) e. On
136	130, 135	syl6eqel 2709	. . . . . . . . . 10 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> y e. On )
137	136	adantl 482	. . . . . . . . 9 |- ( ( A e. V /\ ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) -> y e. On )
138		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> ( r \ _I ) We dom r )
139	133	oien 8443	. . . . . . . . . . . . 13 |- ( ( dom r e. _V /\ ( r \ _I ) We dom r ) -> dom OrdIso ( ( r \ _I ) , dom r ) ~~ dom r )
140	132, 138, 139	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> dom OrdIso ( ( r \ _I ) , dom r ) ~~ dom r )
141	130, 140	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> y ~~ dom r )
142	141	adantl 482	. . . . . . . . . 10 |- ( ( A e. V /\ ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) -> y ~~ dom r )
143		simpll1 1100	. . . . . . . . . . 11 |- ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> dom r C_ A )
144		ssdomg 8001	. . . . . . . . . . . 12 |- ( A e. V -> ( dom r C_ A -> dom r ~<_ A ) )
145	144	imp 445	. . . . . . . . . . 11 |- ( ( A e. V /\ dom r C_ A ) -> dom r ~<_ A )
146	143, 145	sylan2 491	. . . . . . . . . 10 |- ( ( A e. V /\ ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) -> dom r ~<_ A )
147		endomtr 8014	. . . . . . . . . 10 |- ( ( y ~~ dom r /\ dom r ~<_ A ) -> y ~<_ A )
148	142, 146, 147	syl2anc 693	. . . . . . . . 9 |- ( ( A e. V /\ ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) -> y ~<_ A )
149	137, 148	jca 554	. . . . . . . 8 |- ( ( A e. V /\ ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) -> ( y e. On /\ y ~<_ A ) )
150	149	ex 450	. . . . . . 7 |- ( A e. V -> ( ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> ( y e. On /\ y ~<_ A ) ) )
151	150	exlimdv 1861	. . . . . 6 |- ( A e. V -> ( E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) -> ( y e. On /\ y ~<_ A ) ) )
152	129, 151	impbid2 216	. . . . 5 |- ( A e. V -> ( ( y e. On /\ y ~<_ A ) <-> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) )
153	20, 152	syl5bb 272	. . . 4 |- ( A e. V -> ( y e. { x e. On | x ~<_ A } <-> E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) ) )
154	153	abbi2dv 2742	. . 3 |- ( A e. V -> { x e. On | x ~<_ A } = { y | E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
155	1	rneqi 5352	. . . 4 |- ran F = ran { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
156		rnopab 5370	. . . 4 |- ran { <. r , y >. | ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = { y | E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
157	155, 156	eqtri 2644	. . 3 |- ran F = { y | E. r ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
158	154, 157	syl6reqr 2675	. 2 |- ( A e. V -> ran F = { x e. On | x ~<_ A } )
159	15, 18, 158	3pm3.2i 1239	1 |- ( dom F C_ ~P ( A X. A ) /\ Fun F /\ ( A e. V -> ran F = { x e. On | x ~<_ A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   {copab 4712   _I cid 5023   _E cep 5028   Or wor 5034   Se wse 5071   We wwe 5072   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   Ord word 5722   Oncon0 5723   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   ~~ cen 7952   ~<_ cdom 7953  OrdIsocoi 8414

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-iun 4522  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-wrecs 7407  df-recs 7468  df-en 7956  df-dom 7957  df-oi 8415

This theorem is referenced by:  hartogslem2  8448  harwdom  8495