Theorem mapvalg 3263

Description: The value of set exponentiation. (A ^m B) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24.

Assertion

Ref	Expression
mapvalg	|- ((A e. C /\ B e. D) -> (A ^m B) = {f | f:B-->A})

Distinct variable group(s):   A,f   B,f

Proof of Theorem mapvalg

Step	Hyp	Ref	Expression
1		mapex 3261	. . 3 |- ((B e. D /\ A e. C) -> {f | f:B-->A} e. V)
2	1	ancoms 334	. 2 |- ((A e. C /\ B e. D) -> {f | f:B-->A} e. V)
3		feq3 2750	. . . . . . 7 |- (x = A -> (f:y-->x <-> f:y-->A))
4	3	biabdv 1183	. . . . . 6 |- (x = A -> {f | f:y-->x} = {f | f:y-->A})
5		feq2 2749	. . . . . . 7 |- (y = B -> (f:y-->A <-> f:B-->A))
6	5	biabdv 1183	. . . . . 6 |- (y = B -> {f | f:y-->A} = {f | f:B-->A})
7		df-map 3259	. . . . . . 7 |- ^m = {<.<.x, y>., z>. | z = {f | f:y-->x}}
8		visset 1350	. . . . . . . . . 10 |- x e. V
9		visset 1350	. . . . . . . . . 10 |- y e. V
10	8, 9	pm3.2i 234	. . . . . . . . 9 |- (x e. V /\ y e. V)
11	10	biantrur 544	. . . . . . . 8 |- (z = {f | f:y-->x} <-> ((x e. V /\ y e. V) /\ z = {f | f:y-->x}))
12	11	bioprabi 3027	. . . . . . 7 |- {<.<.x, y>., z>. | z = {f | f:y-->x}} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | f:y-->x})}
13	7, 12	eqtr 1119	. . . . . 6 |- ^m = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | f:y-->x})}
14	4, 6, 13	oprabval2g 3050	. . . . 5 |- ((A e. V /\ B e. V /\ {f | f:B-->A} e. V) -> (A ^m B) = {f | f:B-->A})
15	14	3exp 611	. . . 4 |- (A e. V -> (B e. V -> ({f | f:B-->A} e. V -> (A ^m B) = {f | f:B-->A})))
16	15	imp 277	. . 3 |- ((A e. V /\ B e. V) -> ({f | f:B-->A} e. V -> (A ^m B) = {f | f:B-->A}))
17		elisset 1354	. . 3 |- (A e. C -> A e. V)
18		elisset 1354	. . 3 |- (B e. D -> B e. V)
19	16, 17, 18	syl2an 349	. 2 |- ((A e. C /\ B e. D) -> ({f | f:B-->A} e. V -> (A ^m B) = {f | f:B-->A}))
20	2, 19	mpd 46	1 |- ((A e. C /\ B e. D) -> (A ^m B) = {f | f:B-->A})

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  -->wf 2418  (class class class)co 3001  {copab2 3002   ^m cm 3258

This theorem is referenced by:  mapval 3264  mapss 3270

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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