Theorem sofld 5274

Description: The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
sofld	|- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A = ( dom R u. ran R ) )

Proof of Theorem sofld

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 4933	. . . . . . . . 9 |- Rel ( A X. A )
2		relss 4913	. . . . . . . . 9 |- ( R C_ ( A X. A ) -> ( Rel ( A X. A ) -> Rel R ) )
3	1, 2	mpi 21	. . . . . . . 8 |- ( R C_ ( A X. A ) -> Rel R )
4	3	ad2antlr 727	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> Rel R )
5		df-br 4398	. . . . . . . . . 10 |- ( x R y <-> <. x , y >. e. R )
6		ssun1 3608	. . . . . . . . . . . . 13 |- A C_ ( A u. { x } )
7		undif1 3849	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
8	6, 7	sseqtr4i 3477	. . . . . . . . . . . 12 |- A C_ ( ( A \ { x } ) u. { x } )
9		simpll 754	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> R Or A )
10		dmss 5025	. . . . . . . . . . . . . . . . 17 |- ( R C_ ( A X. A ) -> dom R C_ dom ( A X. A ) )
11		dmxpid 5045	. . . . . . . . . . . . . . . . 17 |- dom ( A X. A ) = A
12	10, 11	syl6sseq 3490	. . . . . . . . . . . . . . . 16 |- ( R C_ ( A X. A ) -> dom R C_ A )
13	12	ad2antlr 727	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> dom R C_ A )
14	3	ad2antlr 727	. . . . . . . . . . . . . . . 16 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> Rel R )
15		releldm 5058	. . . . . . . . . . . . . . . 16 |- ( ( Rel R /\ x R y ) -> x e. dom R )
16	14, 15	sylancom 667	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. dom R )
17	13, 16	sseldd 3445	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. A )
18		sossfld 5273	. . . . . . . . . . . . . 14 |- ( ( R Or A /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
19	9, 17, 18	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
20		ssun1 3608	. . . . . . . . . . . . . . 15 |- dom R C_ ( dom R u. ran R )
21	20, 16	sseldi 3442	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. ( dom R u. ran R ) )
22	21	snssd 4119	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> { x } C_ ( dom R u. ran R ) )
23	19, 22	unssd 3621	. . . . . . . . . . . 12 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> ( ( A \ { x } ) u. { x } ) C_ ( dom R u. ran R ) )
24	8, 23	syl5ss 3455	. . . . . . . . . . 11 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> A C_ ( dom R u. ran R ) )
25	24	ex 434	. . . . . . . . . 10 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( x R y -> A C_ ( dom R u. ran R ) ) )
26	5, 25	syl5bir 220	. . . . . . . . 9 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( <. x , y >. e. R -> A C_ ( dom R u. ran R ) ) )
27	26	con3dimp 441	. . . . . . . 8 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> -. <. x , y >. e. R )
28	27	pm2.21d 108	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> ( <. x , y >. e. R -> <. x , y >. e. (/) ) )
29	4, 28	relssdv 4918	. . . . . 6 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R C_ (/) )
30		ss0 3772	. . . . . 6 |- ( R C_ (/) -> R = (/) )
31	29, 30	syl 17	. . . . 5 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R = (/) )
32	31	ex 434	. . . 4 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( -. A C_ ( dom R u. ran R ) -> R = (/) ) )
33	32	necon1ad 2621	. . 3 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( R =/= (/) -> A C_ ( dom R u. ran R ) ) )
34	33	3impia 1196	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A C_ ( dom R u. ran R ) )
35		rnss 5054	. . . . 5 |- ( R C_ ( A X. A ) -> ran R C_ ran ( A X. A ) )
36		rnxpid 5260	. . . . 5 |- ran ( A X. A ) = A
37	35, 36	syl6sseq 3490	. . . 4 |- ( R C_ ( A X. A ) -> ran R C_ A )
38	12, 37	unssd 3621	. . 3 |- ( R C_ ( A X. A ) -> ( dom R u. ran R ) C_ A )
39	38	3ad2ant2 1021	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> ( dom R u. ran R ) C_ A )
40	34, 39	eqssd 3461	1 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A = ( dom R u. ran R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 976   = wceq 1407   e. wcel 1844   =/= wne 2600   \ cdif 3413   u. cun 3414   C_ wss 3416   (/)c0 3740   {csn 3974   <.cop 3980   class class class wbr 4397   Or wor 4745   X. cxp 4823   dom cdm 4825   ran crn 4826   Rel wrel 4830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836

This theorem is referenced by: (None)