Theorem mapdom1g 6944

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 6832	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 4719	. . . . 5 |- ( A ~<_ B -> B e. _V )
3		domeng 6841	. . . . 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 176	. . 3 |- ( A ~<_ B -> E. x ( A ~~ x /\ x C_ B ) )
6	5	adantr 276	. 2 |- ( ( A ~<_ B /\ C e. V ) -> E. x ( A ~~ x /\ x C_ B ) )
7		simpl 109	. . . 4 |- ( ( A ~~ x /\ x C_ B ) -> A ~~ x )
8		enrefg 6855	. . . . 5 |- ( C e. V -> C ~~ C )
9	8	adantl 277	. . . 4 |- ( ( A ~<_ B /\ C e. V ) -> C ~~ C )
10		mapen 6943	. . . 4 |- ( ( A ~~ x /\ C ~~ C ) -> ( A ^m C ) ~~ ( x ^m C ) )
11	7, 9, 10	syl2anr 290	. . 3 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~~ ( x ^m C ) )
12	2	ad2antrr 488	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> B e. _V )
13		simprr 531	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> x C_ B )
14		mapss 6778	. . . . 5 |- ( ( B e. _V /\ x C_ B ) -> ( x ^m C ) C_ ( B ^m C ) )
15	12, 13, 14	syl2anc 411	. . . 4 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) C_ ( B ^m C ) )
16		fnmap 6742	. . . . . . 7 |- ^m Fn ( _V X. _V )
17		elex 2783	. . . . . . 7 |- ( C e. V -> C e. _V )
18		fnovex 5977	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ B e. _V /\ C e. _V ) -> ( B ^m C ) e. _V )
19	16, 2, 17, 18	mp3an3an 1356	. . . . . 6 |- ( ( A ~<_ B /\ C e. V ) -> ( B ^m C ) e. _V )
20		ssdomg 6870	. . . . . 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 276	. . . 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 6882	. . 3 |- ( ( ( A ^m C ) ~~ ( x ^m C ) /\ ( x ^m C ) ~<_ ( B ^m C ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
25	11, 23, 24	syl2anc 411	. 2 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
26	6, 25	exlimddv 1922	1 |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1515   e. wcel 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   X. cxp 4673   Fn wfn 5266  (class class class)co 5944   ^m cmap 6735   ~~ cen 6825   ~<_ cdom 6826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-en 6828  df-dom 6829

This theorem is referenced by: (None)