Theorem mapss 3270

Description: Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89.

Hypotheses

Ref	Expression
mapss.1	|- B e. V
mapss.2	|- C e. V

Assertion

Ref	Expression
mapss	|- (A (_ B -> (A ^m C) (_ (B ^m C))

Proof of Theorem mapss

Step	Hyp	Ref	Expression
1		fss 2759	. . . . . 6 |- ((f:C-->A /\ A (_ B) -> f:C-->B)
2	1	exp 291	. . . . 5 |- (f:C-->A -> (A (_ B -> f:C-->B))
3	2	com12 13	. . . 4 |- (A (_ B -> (f:C-->A -> f:C-->B))
4	3	19.21aiv 943	. . 3 |- (A (_ B -> A.f(f:C-->A -> f:C-->B))
5		ss2ab 1551	. . 3 |- ({f | f:C-->A} (_ {f | f:C-->B} <-> A.f(f:C-->A -> f:C-->B))
6	4, 5	sylibr 175	. 2 |- (A (_ B -> {f | f:C-->A} (_ {f | f:C-->B})
7		mapss.1	. . . 4 |- B e. V
8	7	ssex 1700	. . 3 |- (A (_ B -> A e. V)
9		mapss.2	. . . 4 |- C e. V
10		mapvalg 3263	. . . 4 |- ((A e. V /\ C e. V) -> (A ^m C) = {f | f:C-->A})
11	9, 10	mpan2 519	. . 3 |- (A e. V -> (A ^m C) = {f | f:C-->A})
12	8, 11	syl 12	. 2 |- (A (_ B -> (A ^m C) = {f | f:C-->A})
13	7, 9	mapval 3264	. . 3 |- (B ^m C) = {f | f:C-->B}
14	13	a1i 7	. 2 |- (A (_ B -> (B ^m C) = {f | f:C-->B})
15	6, 12, 14	3sstr4d 1543	1 |- (A (_ B -> (A ^m C) (_ (B ^m C))

Syntax hints:   -> wi 2  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  -->wf 2418  (class class class)co 3001   ^m cm 3258

This theorem is referenced by:  mapdom1 3387

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004  df-map 3259

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