Theorem mapdom1 4492

Description: Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149.

Hypotheses

Ref	Expression
mapdom1.1	|- A e. V
mapdom1.2	|- B e. V
mapdom1.3	|- C e. V

Assertion

Ref	Expression
mapdom1	|- (A ~<_ B -> (A ^m C) ~<_ (B ^m C))

Proof of Theorem mapdom1

Step	Hyp	Ref	Expression
1		mapdom1.2	. . 3 |- B e. V
2	1	domen 4379	. 2 |- (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))
3		endomtr 4420	. . . 4 |- (((A ^m C) ~~ (x ^m C) /\ (x ^m C) ~<_ (B ^m C)) -> (A ^m C) ~<_ (B ^m C))
4		mapdom1.3	. . . . . 6 |- C e. V
5	4	enref 4391	. . . . 5 |- C ~~ C
6		mapdom1.1	. . . . . 6 |- A e. V
7		visset 1813	. . . . . 6 |- x e. V
8	6, 7, 4, 4	mapen 4491	. . . . 5 |- ((A ~~ x /\ C ~~ C) -> (A ^m C) ~~ (x ^m C))
9	5, 8	mpan2 696	. . . 4 |- (A ~~ x -> (A ^m C) ~~ (x ^m C))
10	1, 4	mapss 4346	. . . . 5 |- (x (_ B -> (x ^m C) (_ (B ^m C))
11		oprex 3983	. . . . . 6 |- (x ^m C) e. V
12		ssdomg 4408	. . . . . 6 |- ((x ^m C) e. V -> ((x ^m C) (_ (B ^m C) -> (x ^m C) ~<_ (B ^m C)))
13	11, 12	ax-mp 7	. . . . 5 |- ((x ^m C) (_ (B ^m C) -> (x ^m C) ~<_ (B ^m C))
14	10, 13	syl 10	. . . 4 |- (x (_ B -> (x ^m C) ~<_ (B ^m C))
15	3, 9, 14	syl2an 454	. . 3 |- ((A ~~ x /\ x (_ B) -> (A ^m C) ~<_ (B ^m C))
16	15	19.23aiv 1295	. 2 |- (E.x(A ~~ x /\ x (_ B) -> (A ^m C) ~<_ (B ^m C))
17	2, 16	sylbi 199	1 |- (A ~<_ B -> (A ^m C) ~<_ (B ^m C))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  E.wex 980  Vcvv 1811   (_ wss 2047   class class class wbr 2619  (class class class)co 3963   ^m cm 4322   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  infmap1 7573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-en 4368  df-dom 4369

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