Theorem mapdom2 8131

Description: Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
mapdom2	|- ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) -> ( C ^m A ) ~<_ ( C ^m B ) )

Proof of Theorem mapdom2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . 8 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> C = (/) )
2	1	oveq1d 6665	. . . . . . 7 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( C ^m A ) = ( (/) ^m A ) )
3		simplr 792	. . . . . . . . . 10 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> -. ( A = (/) /\ C = (/) ) )
4		idd 24	. . . . . . . . . . 11 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( A = (/) -> A = (/) ) )
5	4, 1	jctird 567	. . . . . . . . . 10 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( A = (/) -> ( A = (/) /\ C = (/) ) ) )
6	3, 5	mtod 189	. . . . . . . . 9 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> -. A = (/) )
7	6	neqned 2801	. . . . . . . 8 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> A =/= (/) )
8		map0b 7896	. . . . . . . 8 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )
9	7, 8	syl 17	. . . . . . 7 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( (/) ^m A ) = (/) )
10	2, 9	eqtrd 2656	. . . . . 6 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( C ^m A ) = (/) )
11		ovex 6678	. . . . . . 7 |- ( C ^m B ) e. _V
12	11	0dom 8090	. . . . . 6 |- (/) ~<_ ( C ^m B )
13	10, 12	syl6eqbr 4692	. . . . 5 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C = (/) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
14		simpll 790	. . . . . . . 8 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> A ~<_ B )
15		reldom 7961	. . . . . . . . . . 11 |- Rel ~<_
16	15	brrelex2i 5159	. . . . . . . . . 10 |- ( A ~<_ B -> B e. _V )
17	16	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> B e. _V )
18		domeng 7969	. . . . . . . . 9 |- ( B e. _V -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
19	17, 18	syl 17	. . . . . . . 8 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
20	14, 19	mpbid 222	. . . . . . 7 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> E. x ( A ~~ x /\ x C_ B ) )
21		enrefg 7987	. . . . . . . . . . . 12 |- ( C e. _V -> C ~~ C )
22	21	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> C ~~ C )
23		simprrl 804	. . . . . . . . . . 11 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> A ~~ x )
24		mapen 8124	. . . . . . . . . . 11 |- ( ( C ~~ C /\ A ~~ x ) -> ( C ^m A ) ~~ ( C ^m x ) )
25	22, 23, 24	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m A ) ~~ ( C ^m x ) )
26		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m x ) e. _V )
27		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m ( B \ x ) ) e. _V )
28		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> C =/= (/) )
29		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> C e. _V )
30	16	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> B e. _V )
31		difexg 4808	. . . . . . . . . . . . . . 15 |- ( B e. _V -> ( B \ x ) e. _V )
32	30, 31	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( B \ x ) e. _V )
33		map0g 7897	. . . . . . . . . . . . . . . 16 |- ( ( C e. _V /\ ( B \ x ) e. _V ) -> ( ( C ^m ( B \ x ) ) = (/) <-> ( C = (/) /\ ( B \ x ) =/= (/) ) ) )
34		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( C = (/) /\ ( B \ x ) =/= (/) ) -> C = (/) )
35	33, 34	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( ( C e. _V /\ ( B \ x ) e. _V ) -> ( ( C ^m ( B \ x ) ) = (/) -> C = (/) ) )
36	35	necon3d 2815	. . . . . . . . . . . . . 14 |- ( ( C e. _V /\ ( B \ x ) e. _V ) -> ( C =/= (/) -> ( C ^m ( B \ x ) ) =/= (/) ) )
37	29, 32, 36	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C =/= (/) -> ( C ^m ( B \ x ) ) =/= (/) ) )
38	28, 37	mpd 15	. . . . . . . . . . . 12 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m ( B \ x ) ) =/= (/) )
39		xpdom3 8058	. . . . . . . . . . . 12 |- ( ( ( C ^m x ) e. _V /\ ( C ^m ( B \ x ) ) e. _V /\ ( C ^m ( B \ x ) ) =/= (/) ) -> ( C ^m x ) ~<_ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) )
40	26, 27, 38, 39	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m x ) ~<_ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) )
41		vex 3203	. . . . . . . . . . . . . . 15 |- x e. _V
42	41	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> x e. _V )
43		disjdif 4040	. . . . . . . . . . . . . . 15 |- ( x i^i ( B \ x ) ) = (/)
44	43	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( x i^i ( B \ x ) ) = (/) )
45		mapunen 8129	. . . . . . . . . . . . . 14 |- ( ( ( x e. _V /\ ( B \ x ) e. _V /\ C e. _V ) /\ ( x i^i ( B \ x ) ) = (/) ) -> ( C ^m ( x u. ( B \ x ) ) ) ~~ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) )
46	42, 32, 29, 44, 45	syl31anc 1329	. . . . . . . . . . . . 13 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m ( x u. ( B \ x ) ) ) ~~ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) )
47	46	ensymd 8007	. . . . . . . . . . . 12 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) ~~ ( C ^m ( x u. ( B \ x ) ) ) )
48		simprrr 805	. . . . . . . . . . . . . 14 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> x C_ B )
49		undif 4049	. . . . . . . . . . . . . 14 |- ( x C_ B <-> ( x u. ( B \ x ) ) = B )
50	48, 49	sylib 208	. . . . . . . . . . . . 13 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( x u. ( B \ x ) ) = B )
51	50	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m ( x u. ( B \ x ) ) ) = ( C ^m B ) )
52	47, 51	breqtrd 4679	. . . . . . . . . . 11 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) ~~ ( C ^m B ) )
53		domentr 8015	. . . . . . . . . . 11 |- ( ( ( C ^m x ) ~<_ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) /\ ( ( C ^m x ) X. ( C ^m ( B \ x ) ) ) ~~ ( C ^m B ) ) -> ( C ^m x ) ~<_ ( C ^m B ) )
54	40, 52, 53	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m x ) ~<_ ( C ^m B ) )
55		endomtr 8014	. . . . . . . . . 10 |- ( ( ( C ^m A ) ~~ ( C ^m x ) /\ ( C ^m x ) ~<_ ( C ^m B ) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
56	25, 54, 55	syl2anc 693	. . . . . . . . 9 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( C =/= (/) /\ ( A ~~ x /\ x C_ B ) ) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
57	56	expr 643	. . . . . . . 8 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> ( ( A ~~ x /\ x C_ B ) -> ( C ^m A ) ~<_ ( C ^m B ) ) )
58	57	exlimdv 1861	. . . . . . 7 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> ( E. x ( A ~~ x /\ x C_ B ) -> ( C ^m A ) ~<_ ( C ^m B ) ) )
59	20, 58	mpd 15	. . . . . 6 |- ( ( ( A ~<_ B /\ C e. _V ) /\ C =/= (/) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
60	59	adantlr 751	. . . . 5 |- ( ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) /\ C =/= (/) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
61	13, 60	pm2.61dane 2881	. . . 4 |- ( ( ( A ~<_ B /\ C e. _V ) /\ -. ( A = (/) /\ C = (/) ) ) -> ( C ^m A ) ~<_ ( C ^m B ) )
62	61	an32s 846	. . 3 |- ( ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) /\ C e. _V ) -> ( C ^m A ) ~<_ ( C ^m B ) )
63	62	ex 450	. 2 |- ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) -> ( C e. _V -> ( C ^m A ) ~<_ ( C ^m B ) ) )
64		reldmmap 7866	. . . 4 |- Rel dom ^m
65	64	ovprc1 6684	. . 3 |- ( -. C e. _V -> ( C ^m A ) = (/) )
66	65, 12	syl6eqbr 4692	. 2 |- ( -. C e. _V -> ( C ^m A ) ~<_ ( C ^m B ) )
67	63, 66	pm2.61d1 171	1 |- ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) -> ( C ^m A ) ~<_ ( C ^m B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   X. cxp 5112  (class class class)co 6650   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953

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-sbc 3436  df-csb 3534  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-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957

This theorem is referenced by:  mapdom3  8132  cfpwsdom  9406  hauspwdom  21304