Theorem mapdom1g 6865

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 6763	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 4685	. . . . 5 |- ( A ~<_ B -> B e. _V )
3		domeng 6770	. . . . 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 6782	. . . . 5 |- ( C e. V -> C ~~ C )
9	8	adantl 277	. . . 4 |- ( ( A ~<_ B /\ C e. V ) -> C ~~ C )
10		mapen 6864	. . . 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 6709	. . . . 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 6673	. . . . . . 7 |- ^m Fn ( _V X. _V )
17		elex 2763	. . . . . . 7 |- ( C e. V -> C e. _V )
18		fnovex 5924	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ B e. _V /\ C e. _V ) -> ( B ^m C ) e. _V )
19	16, 2, 17, 18	mp3an3an 1354	. . . . . 6 |- ( ( A ~<_ B /\ C e. V ) -> ( B ^m C ) e. _V )
20		ssdomg 6796	. . . . . 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 6808	. . 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 1910	1 |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1503   e. wcel 2160   _Vcvv 2752   C_ wss 3144   class class class wbr 4018   X. cxp 4639   Fn wfn 5226  (class class class)co 5891   ^m cmap 6666   ~~ cen 6756   ~<_ cdom 6757

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-en 6759  df-dom 6760

This theorem is referenced by: (None)