Theorem mapdom1g 7032

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 6913	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 4770	. . . . 5 |- ( A ~<_ B -> B e. _V )
3		domeng 6922	. . . . 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 6936	. . . . 5 |- ( C e. V -> C ~~ C )
9	8	adantl 277	. . . 4 |- ( ( A ~<_ B /\ C e. V ) -> C ~~ C )
10		mapen 7031	. . . 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 533	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> x C_ B )
14		mapss 6859	. . . . 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 6823	. . . . . . 7 |- ^m Fn ( _V X. _V )
17		elex 2814	. . . . . . 7 |- ( C e. V -> C e. _V )
18		fnovex 6050	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ B e. _V /\ C e. _V ) -> ( B ^m C ) e. _V )
19	16, 2, 17, 18	mp3an3an 1379	. . . . . 6 |- ( ( A ~<_ B /\ C e. V ) -> ( B ^m C ) e. _V )
20		ssdomg 6951	. . . . . 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 6963	. . 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 1947	1 |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1540   e. wcel 2202   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   X. cxp 4723   Fn wfn 5321  (class class class)co 6017   ^m cmap 6816   ~~ cen 6906   ~<_ cdom 6907

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-en 6909  df-dom 6910

This theorem is referenced by: (None)