Theorem mapdom1 8125

Description: Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
mapdom1	|- ( A ~<_ B -> ( A ^m C ) ~<_ ( B ^m C ) )

Proof of Theorem mapdom1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . . . 7 |- Rel ~<_
2	1	brrelex2i 5159	. . . . . 6 |- ( A ~<_ B -> B e. _V )
3		domeng 7969	. . . . . 6 |- ( B e. _V -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
4	2, 3	syl 17	. . . . 5 |- ( A ~<_ B -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
5	4	ibi 256	. . . 4 |- ( A ~<_ B -> E. x ( A ~~ x /\ x C_ B ) )
6	5	adantr 481	. . 3 |- ( ( A ~<_ B /\ C e. _V ) -> E. x ( A ~~ x /\ x C_ B ) )
7		simpl 473	. . . . 5 |- ( ( A ~~ x /\ x C_ B ) -> A ~~ x )
8		enrefg 7987	. . . . . 6 |- ( C e. _V -> C ~~ C )
9	8	adantl 482	. . . . 5 |- ( ( A ~<_ B /\ C e. _V ) -> C ~~ C )
10		mapen 8124	. . . . 5 |- ( ( A ~~ x /\ C ~~ C ) -> ( A ^m C ) ~~ ( x ^m C ) )
11	7, 9, 10	syl2anr 495	. . . 4 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~~ ( x ^m C ) )
12		ovex 6678	. . . . 5 |- ( B ^m C ) e. _V
13	2	ad2antrr 762	. . . . . 6 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> B e. _V )
14		simprr 796	. . . . . 6 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> x C_ B )
15		mapss 7900	. . . . . 6 |- ( ( B e. _V /\ x C_ B ) -> ( x ^m C ) C_ ( B ^m C ) )
16	13, 14, 15	syl2anc 693	. . . . 5 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) C_ ( B ^m C ) )
17		ssdomg 8001	. . . . 5 |- ( ( B ^m C ) e. _V -> ( ( x ^m C ) C_ ( B ^m C ) -> ( x ^m C ) ~<_ ( B ^m C ) ) )
18	12, 16, 17	mpsyl 68	. . . 4 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) ~<_ ( B ^m C ) )
19		endomtr 8014	. . . 4 |- ( ( ( A ^m C ) ~~ ( x ^m C ) /\ ( x ^m C ) ~<_ ( B ^m C ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
20	11, 18, 19	syl2anc 693	. . 3 |- ( ( ( A ~<_ B /\ C e. _V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
21	6, 20	exlimddv 1863	. 2 |- ( ( A ~<_ B /\ C e. _V ) -> ( A ^m C ) ~<_ ( B ^m C ) )
22		elmapex 7878	. . . . . . 7 |- ( x e. ( A ^m C ) -> ( A e. _V /\ C e. _V ) )
23	22	simprd 479	. . . . . 6 |- ( x e. ( A ^m C ) -> C e. _V )
24	23	con3i 150	. . . . 5 |- ( -. C e. _V -> -. x e. ( A ^m C ) )
25	24	eq0rdv 3979	. . . 4 |- ( -. C e. _V -> ( A ^m C ) = (/) )
26	25	adantl 482	. . 3 |- ( ( A ~<_ B /\ -. C e. _V ) -> ( A ^m C ) = (/) )
27	12	0dom 8090	. . 3 |- (/) ~<_ ( B ^m C )
28	26, 27	syl6eqbr 4692	. 2 |- ( ( A ~<_ B /\ -. C e. _V ) -> ( A ^m C ) ~<_ ( B ^m C ) )
29	21, 28	pm2.61dan 832	1 |- ( A ~<_ B -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (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-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-map 7859  df-en 7956  df-dom 7957

This theorem is referenced by:  mappwen  8935  pwcfsdom  9405  cfpwsdom  9406  rpnnen  14956  rexpen  14957  hauspwdom  21304