Theorem mapss 4339

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 3630	. . . . 5 |- ((f:C-->A /\ A (_ B) -> f:C-->B)
2	1	expcom 374	. . . 4 |- (A (_ B -> (f:C-->A -> f:C-->B))
3	2	19.21aiv 1285	. . 3 |- (A (_ B -> A.f(f:C-->A -> f:C-->B))
4		ss2ab 2113	. . 3 |- ({f | f:C-->A} (_ {f | f:C-->B} <-> A.f(f:C-->A -> f:C-->B))
5	3, 4	sylibr 200	. 2 |- (A (_ B -> {f | f:C-->A} (_ {f | f:C-->B})
6		mapss.1	. . . 4 |- B e. V
7	6	ssex 2715	. . 3 |- (A (_ B -> A e. V)
8		mapss.2	. . . 4 |- C e. V
9		mapvalg 4323	. . . 4 |- ((A e. V /\ C e. V) -> (A ^m C) = {f | f:C-->A})
10	8, 9	mpan2 695	. . 3 |- (A e. V -> (A ^m C) = {f | f:C-->A})
11	7, 10	syl 10	. 2 |- (A (_ B -> (A ^m C) = {f | f:C-->A})
12	6, 8	mapval 4325	. . 3 |- (B ^m C) = {f | f:C-->B}
13	12	a1i 8	. 2 |- (A (_ B -> (B ^m C) = {f | f:C-->B})
14	5, 11, 13	3sstr4d 2101	1 |- (A (_ B -> (A ^m C) (_ (B ^m C))

Syntax hints:   -> wi 3  A.wal 953   = wceq 955   e. wcel 957  {cab 1462  Vcvv 1808   (_ wss 2044  -->wf 3174  (class class class)co 3958   ^m cm 4315

This theorem is referenced by:  mapdom1 4481

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-oprab 3961  df-map 4317

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