Theorem sofld 5448

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 5103	. . . . . . . . 9 |- Rel ( A X. A )
2		relss 5083	. . . . . . . . 9 |- ( R C_ ( A X. A ) -> ( Rel ( A X. A ) -> Rel R ) )
3	1, 2	mpi 17	. . . . . . . 8 |- ( R C_ ( A X. A ) -> Rel R )
4	3	ad2antlr 726	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> Rel R )
5		df-br 4443	. . . . . . . . . 10 |- ( x R y <-> <. x , y >. e. R )
6		ssun1 3662	. . . . . . . . . . . . 13 |- A C_ ( A u. { x } )
7		undif1 3897	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
8	6, 7	sseqtr4i 3532	. . . . . . . . . . . 12 |- A C_ ( ( A \ { x } ) u. { x } )
9		simpll 753	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> R Or A )
10		dmss 5195	. . . . . . . . . . . . . . . . 17 |- ( R C_ ( A X. A ) -> dom R C_ dom ( A X. A ) )
11		dmxpid 5215	. . . . . . . . . . . . . . . . 17 |- dom ( A X. A ) = A
12	10, 11	syl6sseq 3545	. . . . . . . . . . . . . . . 16 |- ( R C_ ( A X. A ) -> dom R C_ A )
13	12	ad2antlr 726	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> dom R C_ A )
14	3	ad2antlr 726	. . . . . . . . . . . . . . . 16 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> Rel R )
15		releldm 5228	. . . . . . . . . . . . . . . 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 3500	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. A )
18		sossfld 5447	. . . . . . . . . . . . . 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 3662	. . . . . . . . . . . . . . 15 |- dom R C_ ( dom R u. ran R )
21	20, 16	sseldi 3497	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. ( dom R u. ran R ) )
22	21	snssd 4167	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> { x } C_ ( dom R u. ran R ) )
23	19, 22	unssd 3675	. . . . . . . . . . . 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 3510	. . . . . . . . . . 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 218	. . . . . . . . 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 106	. . . . . . 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 5088	. . . . . 6 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R C_ (/) )
30		ss0 3811	. . . . . 6 |- ( R C_ (/) -> R = (/) )
31	29, 30	syl 16	. . . . 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 2678	. . 3 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( R =/= (/) -> A C_ ( dom R u. ran R ) ) )
34	33	3impia 1188	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A C_ ( dom R u. ran R ) )
35		rnss 5224	. . . . 5 |- ( R C_ ( A X. A ) -> ran R C_ ran ( A X. A ) )
36		rnxpid 5433	. . . . 5 |- ran ( A X. A ) = A
37	35, 36	syl6sseq 3545	. . . 4 |- ( R C_ ( A X. A ) -> ran R C_ A )
38	12, 37	unssd 3675	. . 3 |- ( R C_ ( A X. A ) -> ( dom R u. ran R ) C_ A )
39	38	3ad2ant2 1013	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> ( dom R u. ran R ) C_ A )
40	34, 39	eqssd 3516	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 968   = wceq 1374   e. wcel 1762   =/= wne 2657   \ cdif 3468   u. cun 3469   C_ wss 3471   (/)c0 3780   {csn 4022   <.cop 4028   class class class wbr 4442   Or wor 4794   X. cxp 4992   dom cdm 4994   ran crn 4995   Rel wrel 4999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005

This theorem is referenced by: (None)