Theorem sofld 5290

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 4947	. . . . . . . . 9 |- Rel ( A X. A )
2		relss 4927	. . . . . . . . 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 741	. . . . . . 7 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> Rel R )
5		df-br 4396	. . . . . . . . . 10 |- ( x R y <-> <. x , y >. e. R )
6		ssun1 3588	. . . . . . . . . . . . 13 |- A C_ ( A u. { x } )
7		undif1 3833	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
8	6, 7	sseqtr4i 3451	. . . . . . . . . . . 12 |- A C_ ( ( A \ { x } ) u. { x } )
9		simpll 768	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> R Or A )
10		dmss 5039	. . . . . . . . . . . . . . . . 17 |- ( R C_ ( A X. A ) -> dom R C_ dom ( A X. A ) )
11		dmxpid 5060	. . . . . . . . . . . . . . . . 17 |- dom ( A X. A ) = A
12	10, 11	syl6sseq 3464	. . . . . . . . . . . . . . . 16 |- ( R C_ ( A X. A ) -> dom R C_ A )
13	12	ad2antlr 741	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> dom R C_ A )
14	3	ad2antlr 741	. . . . . . . . . . . . . . . 16 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> Rel R )
15		releldm 5073	. . . . . . . . . . . . . . . 16 |- ( ( Rel R /\ x R y ) -> x e. dom R )
16	14, 15	sylancom 680	. . . . . . . . . . . . . . 15 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. dom R )
17	13, 16	sseldd 3419	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. A )
18		sossfld 5289	. . . . . . . . . . . . . 14 |- ( ( R Or A /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
19	9, 17, 18	syl2anc 673	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
20		ssun1 3588	. . . . . . . . . . . . . . 15 |- dom R C_ ( dom R u. ran R )
21	20, 16	sseldi 3416	. . . . . . . . . . . . . 14 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> x e. ( dom R u. ran R ) )
22	21	snssd 4108	. . . . . . . . . . . . 13 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> { x } C_ ( dom R u. ran R ) )
23	19, 22	unssd 3601	. . . . . . . . . . . 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 3429	. . . . . . . . . . 11 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ x R y ) -> A C_ ( dom R u. ran R ) )
25	24	ex 441	. . . . . . . . . 10 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( x R y -> A C_ ( dom R u. ran R ) ) )
26	5, 25	syl5bir 226	. . . . . . . . 9 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( <. x , y >. e. R -> A C_ ( dom R u. ran R ) ) )
27	26	con3dimp 448	. . . . . . . 8 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> -. <. x , y >. e. R )
28	27	pm2.21d 109	. . . . . . 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 4932	. . . . . 6 |- ( ( ( R Or A /\ R C_ ( A X. A ) ) /\ -. A C_ ( dom R u. ran R ) ) -> R C_ (/) )
30		ss0 3768	. . . . . 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 441	. . . 4 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( -. A C_ ( dom R u. ran R ) -> R = (/) ) )
33	32	necon1ad 2660	. . 3 |- ( ( R Or A /\ R C_ ( A X. A ) ) -> ( R =/= (/) -> A C_ ( dom R u. ran R ) ) )
34	33	3impia 1228	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A C_ ( dom R u. ran R ) )
35		rnss 5069	. . . . 5 |- ( R C_ ( A X. A ) -> ran R C_ ran ( A X. A ) )
36		rnxpid 5276	. . . . 5 |- ran ( A X. A ) = A
37	35, 36	syl6sseq 3464	. . . 4 |- ( R C_ ( A X. A ) -> ran R C_ A )
38	12, 37	unssd 3601	. . 3 |- ( R C_ ( A X. A ) -> ( dom R u. ran R ) C_ A )
39	38	3ad2ant2 1052	. 2 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> ( dom R u. ran R ) C_ A )
40	34, 39	eqssd 3435	1 |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A = ( dom R u. ran R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   u. cun 3388   C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   class class class wbr 4395   Or wor 4759   X. cxp 4837   dom cdm 4839   ran crn 4840   Rel wrel 4844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850

This theorem is referenced by: (None)