Theorem mapdom1g 7137

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 7017	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 4814	. . . . 5 |- ( A ~<_ B -> B e. _V )
3		domeng 7026	. . . . 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 7040	. . . . 5 |- ( C e. V -> C ~~ C )
9	8	adantl 277	. . . 4 |- ( ( A ~<_ B /\ C e. V ) -> C ~~ C )
10		mapen 7136	. . . 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 492	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> B e. _V )
13		simprr 537	. . . . 5 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> x C_ B )
14		mapss 6963	. . . . 5 |- ( ( B e. _V /\ x C_ B ) -> ( x ^m C ) C_ ( B ^m C ) )
15	12, 13, 14	syl2anc 415	. . . 4 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( x ^m C ) C_ ( B ^m C ) )
16		fnmap 6919	. . . . . . 7 |- ^m Fn ( _V X. _V )
17		elex 2833	. . . . . . 7 |- ( C e. V -> C e. _V )
18		fnovex 6108	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ B e. _V /\ C e. _V ) -> ( B ^m C ) e. _V )
19	16, 2, 17, 18	mp3an3an 1384	. . . . . 6 |- ( ( A ~<_ B /\ C e. V ) -> ( B ^m C ) e. _V )
20		ssdomg 7055	. . . . . 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 7067	. . 3 |- ( ( ( A ^m C ) ~~ ( x ^m C ) /\ ( x ^m C ) ~<_ ( B ^m C ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
25	11, 23, 24	syl2anc 415	. 2 |- ( ( ( A ~<_ B /\ C e. V ) /\ ( A ~~ x /\ x C_ B ) ) -> ( A ^m C ) ~<_ ( B ^m C ) )
26	6, 25	exlimddv 1954	1 |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   X. cxp 4767   Fn wfn 5367  (class class class)co 6075   ^m cmap 6912   ~~ cen 7010   ~<_ cdom 7011

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-en 7013  df-dom 7014

This theorem is referenced by: (None)