Theorem mapss 5405

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 C_ B -> (A ^m C) C_ (B ^m C))

Proof of Theorem mapss

Step	Hyp	Ref	Expression
1		fss 4571	. . . 4 |- ((f:C-->A /\ A C_ B) -> f:C-->B)
2	1	expcom 403	. . 3 |- (A C_ B -> (f:C-->A -> f:C-->B))
3	2	ss2abdv 2680	. 2 |- (A C_ B -> {f | f:C-->A} C_ {f | f:C-->B})
4		mapvalg 5389	. . 3 |- ((A e. _V /\ C e. _V) -> (A ^m C) = {f | f:C-->A})
5		mapss.1	. . . 4 |- B e. _V
6	5	ssex 3455	. . 3 |- (A C_ B -> A e. _V)
7		mapss.2	. . 3 |- C e. _V
8	4, 6, 7	sylancl 525	. 2 |- (A C_ B -> (A ^m C) = {f | f:C-->A})
9	5, 7	mapval 5391	. . 3 |- (B ^m C) = {f | f:C-->B}
10	9	a1i 8	. 2 |- (A C_ B -> (B ^m C) = {f | f:C-->B})
11	3, 8, 10	3sstr4d 2660	1 |- (A C_ B -> (A ^m C) C_ (B ^m C))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  -->wf 3994  (class class class)co 4884   ^m cmap 5381

This theorem is referenced by:  mapdom1 5586

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383

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