Theorem domdifsn 8043

Description: Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
domdifsn	|- ( A ~< B -> A ~<_ ( B \ { C } ) )

Proof of Theorem domdifsn

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

Step	Hyp	Ref	Expression
1		sdomdom 7983	. . . . 5 |- ( A ~< B -> A ~<_ B )
2		relsdom 7962	. . . . . . 7 |- Rel ~<
3	2	brrelex2i 5159	. . . . . 6 |- ( A ~< B -> B e. _V )
4		brdomg 7965	. . . . . 6 |- ( B e. _V -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
5	3, 4	syl 17	. . . . 5 |- ( A ~< B -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
6	1, 5	mpbid 222	. . . 4 |- ( A ~< B -> E. f f : A -1-1-> B )
7	6	adantr 481	. . 3 |- ( ( A ~< B /\ C e. B ) -> E. f f : A -1-1-> B )
8		f1f 6101	. . . . . . . 8 |- ( f : A -1-1-> B -> f : A --> B )
9		frn 6053	. . . . . . . 8 |- ( f : A --> B -> ran f C_ B )
10	8, 9	syl 17	. . . . . . 7 |- ( f : A -1-1-> B -> ran f C_ B )
11	10	adantl 482	. . . . . 6 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ran f C_ B )
12		sdomnen 7984	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
13	12	ad2antrr 762	. . . . . . 7 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> -. A ~~ B )
14		vex 3203	. . . . . . . . . . 11 |- f e. _V
15		dff1o5 6146	. . . . . . . . . . . 12 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
16	15	biimpri 218	. . . . . . . . . . 11 |- ( ( f : A -1-1-> B /\ ran f = B ) -> f : A -1-1-onto-> B )
17		f1oen3g 7971	. . . . . . . . . . 11 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
18	14, 16, 17	sylancr 695	. . . . . . . . . 10 |- ( ( f : A -1-1-> B /\ ran f = B ) -> A ~~ B )
19	18	ex 450	. . . . . . . . 9 |- ( f : A -1-1-> B -> ( ran f = B -> A ~~ B ) )
20	19	necon3bd 2808	. . . . . . . 8 |- ( f : A -1-1-> B -> ( -. A ~~ B -> ran f =/= B ) )
21	20	adantl 482	. . . . . . 7 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( -. A ~~ B -> ran f =/= B ) )
22	13, 21	mpd 15	. . . . . 6 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ran f =/= B )
23		pssdifn0 3944	. . . . . 6 |- ( ( ran f C_ B /\ ran f =/= B ) -> ( B \ ran f ) =/= (/) )
24	11, 22, 23	syl2anc 693	. . . . 5 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( B \ ran f ) =/= (/) )
25		n0 3931	. . . . 5 |- ( ( B \ ran f ) =/= (/) <-> E. x x e. ( B \ ran f ) )
26	24, 25	sylib 208	. . . 4 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> E. x x e. ( B \ ran f ) )
27	2	brrelexi 5158	. . . . . . . . 9 |- ( A ~< B -> A e. _V )
28	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A e. _V )
29	3	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> B e. _V )
30		difexg 4808	. . . . . . . . 9 |- ( B e. _V -> ( B \ { x } ) e. _V )
31	29, 30	syl 17	. . . . . . . 8 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> ( B \ { x } ) e. _V )
32		eldifn 3733	. . . . . . . . . . . . 13 |- ( x e. ( B \ ran f ) -> -. x e. ran f )
33		disjsn 4246	. . . . . . . . . . . . 13 |- ( ( ran f i^i { x } ) = (/) <-> -. x e. ran f )
34	32, 33	sylibr 224	. . . . . . . . . . . 12 |- ( x e. ( B \ ran f ) -> ( ran f i^i { x } ) = (/) )
35	34	adantl 482	. . . . . . . . . . 11 |- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ( ran f i^i { x } ) = (/) )
36	10	adantr 481	. . . . . . . . . . . 12 |- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ran f C_ B )
37		reldisj 4020	. . . . . . . . . . . 12 |- ( ran f C_ B -> ( ( ran f i^i { x } ) = (/) <-> ran f C_ ( B \ { x } ) ) )
38	36, 37	syl 17	. . . . . . . . . . 11 |- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ( ( ran f i^i { x } ) = (/) <-> ran f C_ ( B \ { x } ) ) )
39	35, 38	mpbid 222	. . . . . . . . . 10 |- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ran f C_ ( B \ { x } ) )
40		f1ssr 6107	. . . . . . . . . 10 |- ( ( f : A -1-1-> B /\ ran f C_ ( B \ { x } ) ) -> f : A -1-1-> ( B \ { x } ) )
41	39, 40	syldan 487	. . . . . . . . 9 |- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> f : A -1-1-> ( B \ { x } ) )
42	41	adantl 482	. . . . . . . 8 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> f : A -1-1-> ( B \ { x } ) )
43		f1dom2g 7973	. . . . . . . 8 |- ( ( A e. _V /\ ( B \ { x } ) e. _V /\ f : A -1-1-> ( B \ { x } ) ) -> A ~<_ ( B \ { x } ) )
44	28, 31, 42, 43	syl3anc 1326	. . . . . . 7 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A ~<_ ( B \ { x } ) )
45		eldifi 3732	. . . . . . . . 9 |- ( x e. ( B \ ran f ) -> x e. B )
46	45	ad2antll 765	. . . . . . . 8 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> x e. B )
47		simplr 792	. . . . . . . 8 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> C e. B )
48		difsnen 8042	. . . . . . . 8 |- ( ( B e. _V /\ x e. B /\ C e. B ) -> ( B \ { x } ) ~~ ( B \ { C } ) )
49	29, 46, 47, 48	syl3anc 1326	. . . . . . 7 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> ( B \ { x } ) ~~ ( B \ { C } ) )
50		domentr 8015	. . . . . . 7 |- ( ( A ~<_ ( B \ { x } ) /\ ( B \ { x } ) ~~ ( B \ { C } ) ) -> A ~<_ ( B \ { C } ) )
51	44, 49, 50	syl2anc 693	. . . . . 6 |- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A ~<_ ( B \ { C } ) )
52	51	expr 643	. . . . 5 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( x e. ( B \ ran f ) -> A ~<_ ( B \ { C } ) ) )
53	52	exlimdv 1861	. . . 4 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( E. x x e. ( B \ ran f ) -> A ~<_ ( B \ { C } ) ) )
54	26, 53	mpd 15	. . 3 |- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> A ~<_ ( B \ { C } ) )
55	7, 54	exlimddv 1863	. 2 |- ( ( A ~< B /\ C e. B ) -> A ~<_ ( B \ { C } ) )
56	1	adantr 481	. . 3 |- ( ( A ~< B /\ -. C e. B ) -> A ~<_ B )
57		difsn 4328	. . . . 5 |- ( -. C e. B -> ( B \ { C } ) = B )
58	57	breq2d 4665	. . . 4 |- ( -. C e. B -> ( A ~<_ ( B \ { C } ) <-> A ~<_ B ) )
59	58	adantl 482	. . 3 |- ( ( A ~< B /\ -. C e. B ) -> ( A ~<_ ( B \ { C } ) <-> A ~<_ B ) )
60	56, 59	mpbird 247	. 2 |- ( ( A ~< B /\ -. C e. B ) -> A ~<_ ( B \ { C } ) )
61	55, 60	pm2.61dan 832	1 |- ( A ~< B -> A ~<_ ( B \ { C } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  domunsn  8110  marypha1lem  8339