Theorem sofld 5581

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 5227	. . . . . . . . 9 |- Rel ( A X. A )
2		relss 5206	. . . . . . . . 9 |- ( R C_ ( A X. A ) -> ( Rel ( A X. A ) -> Rel R ) )
3	1, 2	mpi 20	. . . . . . . 8 |- ( R C_ ( A X. A ) -> Rel R )
4	3	ad2antlr 763	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> Rel R )
5		df-br 4654	. . . . . . . . . 10 |- ( x R y <-> <. x , y >. e. R )
6		ssun1 3776	. . . . . . . . . . . . 13 |- A C_ ( A u. { x } )
7		undif1 4043	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
8	6, 7	sseqtr4i 3638	. . . . . . . . . . . 12 |- A C_ ( ( A \ { x } ) u. { x } )
9		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> R Or A )
10		dmss 5323	. . . . . . . . . . . . . . . . 17 |- ( R C_ ( A X. A ) -> dom R C_ dom ( A X. A ) )
11		dmxpid 5345	. . . . . . . . . . . . . . . . 17 |- dom ( A X. A ) = A
12	10, 11	syl6sseq 3651	. . . . . . . . . . . . . . . 16 |- ( R C_ ( A X. A ) -> dom R C_ A )
13	12	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> dom R C_ A )
14	3	ad2antlr 763	. . . . . . . . . . . . . . . 16 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> Rel R )
15		releldm 5358	. . . . . . . . . . . . . . . 16 |- ( ( Rel R /\ x R y ) -> x e. dom R )
16	14, 15	sylancom 701	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. dom R )
17	13, 16	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. A )
18		sossfld 5580	. . . . . . . . . . . . . 14 |- ( ( R Or A /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
19	9, 17, 18	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
20		ssun1 3776	. . . . . . . . . . . . . . 15 |- dom R C_ ( dom R u. ran R )
21	20, 16	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. ( dom R u. ran R ) )
22	21	snssd 4340	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> { x } C_ ( dom R u. ran R ) )
23	19, 22	unssd 3789	. . . . . . . . . . . 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 3614	. . . . . . . . . . 11 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> A C_ ( dom R u. ran R ) )
25	24	ex 450	. . . . . . . . . 10 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( x R y -> A C_ ( dom R u. ran R ) ) )
26	5, 25	syl5bir 233	. . . . . . . . 9 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( <. x , y >. e. R -> A C_ ( dom R u. ran R ) ) )
27	26	con3dimp 457	. . . . . . . 8 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> -. <. x , y >. e. R )
28	27	pm2.21d 118	. . . . . . 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 5212	. . . . . 6 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R C_ (/) )
30		ss0 3974	. . . . . 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 450	. . . 4 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( -. A C_ ( dom R u. ran R ) -> R = (/) ) )
33	32	necon1ad 2811	. . 3 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( R =/= (/) -> A C_ ( dom R u. ran R ) ) )
34	33	3impia 1261	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A C_ ( dom R u. ran R ) )
35		rnss 5354	. . . . 5 |- ( R C_ ( A X. A ) -> ran R C_ ran ( A X. A ) )
36		rnxpid 5567	. . . . 5 |- ran ( A X. A ) = A
37	35, 36	syl6sseq 3651	. . . 4 |- ( R C_ ( A X. A ) -> ran R C_ A )
38	12, 37	unssd 3789	. . 3 |- ( R C_ ( A X. A ) -> ( dom R u. ran R ) C_ A )
39	38	3ad2ant2 1083	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> ( dom R u. ran R ) C_ A )
40	34, 39	eqssd 3620	1 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A = ( dom R u. ran R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   Or wor 5034   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)