Theorem sofld 5283

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 4943	. . . . . . . . 9 |- Rel ( A X. A )
2		relss 4923	. . . . . . . . 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 721	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> Rel R )
5		df-br 4290	. . . . . . . . . 10 |- ( x R y <-> <. x , y >. e. R )
6		ssun1 3516	. . . . . . . . . . . . 13 |- A C_ ( A u. { x } )
7		undif1 3751	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
8	6, 7	sseqtr4i 3386	. . . . . . . . . . . 12 |- A C_ ( ( A \ { x } ) u. { x } )
9		simpll 748	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> R Or A )
10		dmss 5035	. . . . . . . . . . . . . . . . 17 |- ( R C_ ( A X. A ) -> dom R C_ dom ( A X. A ) )
11		dmxpid 5055	. . . . . . . . . . . . . . . . 17 |- dom ( A X. A ) = A
12	10, 11	syl6sseq 3399	. . . . . . . . . . . . . . . 16 |- ( R C_ ( A X. A ) -> dom R C_ A )
13	12	ad2antlr 721	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> dom R C_ A )
14	3	ad2antlr 721	. . . . . . . . . . . . . . . 16 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> Rel R )
15		releldm 5068	. . . . . . . . . . . . . . . 16 |- ( ( Rel R /\ x R y ) -> x e. dom R )
16	14, 15	sylancom 662	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. dom R )
17	13, 16	sseldd 3354	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. A )
18		sossfld 5282	. . . . . . . . . . . . . 14 |- ( ( R Or A /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
19	9, 17, 18	syl2anc 656	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
20		ssun1 3516	. . . . . . . . . . . . . . 15 |- dom R C_ ( dom R u. ran R )
21	20, 16	sseldi 3351	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. ( dom R u. ran R ) )
22	21	snssd 4015	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> { x } C_ ( dom R u. ran R ) )
23	19, 22	unssd 3529	. . . . . . . . . . . 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 3364	. . . . . . . . . . 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	con3and 439	. . . . . . . 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 4928	. . . . . 6 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R C_ (/) )
30		ss0 3665	. . . . . 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 2676	. . 3 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( R =/= (/) -> A C_ ( dom R u. ran R ) ) )
34	33	3impia 1179	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A C_ ( dom R u. ran R ) )
35		rnss 5064	. . . . 5 |- ( R C_ ( A X. A ) -> ran R C_ ran ( A X. A ) )
36		rnxpid 5268	. . . . 5 |- ran ( A X. A ) = A
37	35, 36	syl6sseq 3399	. . . 4 |- ( R C_ ( A X. A ) -> ran R C_ A )
38	12, 37	unssd 3529	. . 3 |- ( R C_ ( A X. A ) -> ( dom R u. ran R ) C_ A )
39	38	3ad2ant2 1005	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> ( dom R u. ran R ) C_ A )
40	34, 39	eqssd 3370	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 960   = wceq 1364   e. wcel 1761   =/= wne 2604   \ cdif 3322   u. cun 3323   C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880   class class class wbr 4289   Or wor 4636   X. cxp 4834   dom cdm 4836   ran crn 4837   Rel wrel 4841

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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847

This theorem is referenced by: (None)