Theorem mapdom1g 6822

Description: Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.)

Assertion

Ref	Expression
mapdom1g	|- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Proof of Theorem mapdom1g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 6720	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 4653	. . . . 5 |- ( A ~<_ B -> B e. _V )
3		domeng 6727	. . . . 5 |- ( B e. _V -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
4	2, 3	syl 14	. . . 4 |- ( A ~<_ B -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
5	4	ibi 175	. . 3 |- ( A ~<_ B -> E. x ( A ~~ x /\ x C_ B ) )
6	5	adantr 274	. 2 |- ( ( A ~<_ B /\ C e. V ) -> E. x ( A ~~ x /\ x C_ B ) )
7		simpl 108	. . . 4 |- ( ( A ~~ x /\ x C_ B ) -> A ~~ x )
8		enrefg 6739	. . . . 5 |- ( C e. V -> C ~~ C )
9	8	adantl 275	. . . 4 |- ( ( A ~<_ B /\ C e. V ) -> C ~~ C )
10		mapen 6821	. . . 4 |- ( ( A ~~ x /\ C ~~ C ) -> ( A ^m C ) ~~ ( x ^m C ) )
11	7, 9, 10	syl2anr 288	. . 3 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~~ ( x ^m C ) )
12	2	ad2antrr 485	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> B e. _V )
13		simprr 527	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> x C_ B )
14		mapss 6666	. . . . 5 |- ( ( B e. _V /\ x C_ B ) -> ( x ^m C ) C_ ( B ^m C ) )
15	12, 13, 14	syl2anc 409	. . . 4 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) C_ ( B ^m C ) )
16		fnmap 6630	. . . . . . 7 |- ^m Fn ( _V X. _V )
17		elex 2741	. . . . . . 7 |- ( C e. V -> C e. _V )
18		fnovex 5884	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ B e. _V /\ C e. _V ) -> ( B ^m C ) e. _V )
19	16, 2, 17, 18	mp3an3an 1338	. . . . . 6 |- ( ( A ~<_ B /\ C e. V ) -> ( B ^m C ) e. _V )
20		ssdomg 6753	. . . . . 6 |- ( ( B ^m C ) e. _V -> ( ( x ^m C ) C_ ( B ^m C ) -> ( x ^m C ) ~<_ ( B ^m C ) ) )
21	19, 20	syl 14	. . . . 5 |- ( ( A ~<_ B /\ C e. V ) -> ( ( x ^m C ) C_ ( B ^m C ) -> ( x ^m C ) ~<_ ( B ^m C ) ) )
22	21	adantr 274	. . . 4 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( ( x ^m C ) C_ ( B ^m C ) -> ( x ^m C ) ~<_ ( B ^m C ) ) )
23	15, 22	mpd 13	. . 3 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) ~<_ ( B ^m C ) )
24		endomtr 6765	. . 3 |- ( ( ( A ^m C ) ~~ ( x ^m C ) /\ ( x ^m C ) ~<_ ( B ^m C ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
25	11, 23, 24	syl2anc 409	. 2 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
26	6, 25	exlimddv 1891	1 |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1485   e. wcel 2141   _Vcvv 2730   C_ wss 3121   class class class wbr 3987   X. cxp 4607   Fn wfn 5191  (class class class)co 5851   ^m cmap 6623   ~~ cen 6713   ~<_ cdom 6714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6625  df-en 6716  df-dom 6717

This theorem is referenced by: (None)