Theorem mapvalg 4323

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 groups:   A,f   B,f

Proof of Theorem mapvalg

Step	Hyp	Ref	Expression
1		mapex 4321	. . 3 |- ((B e. D /\ A e. C) -> {f | f:B-->A} e. V)
2	1	ancoms 436	. 2 |- ((A e. C /\ B e. D) -> {f | f:B-->A} e. V)
3		feq3 3618	. . . . . 6 |- (x = A -> (f:y-->x <-> f:y-->A))
4	3	abbidv 1575	. . . . 5 |- (x = A -> {f | f:y-->x} = {f | f:y-->A})
5		feq2 3617	. . . . . 6 |- (y = B -> (f:y-->A <-> f:B-->A))
6	5	abbidv 1575	. . . . 5 |- (y = B -> {f | f:y-->A} = {f | f:B-->A})
7		df-map 4317	. . . . . 6 |- ^m = {<.<.x, y>., z>. | z = {f | f:y-->x}}
8		visset 1810	. . . . . . . . 9 |- x e. V
9		visset 1810	. . . . . . . . 9 |- y e. V
10	8, 9	pm3.2i 285	. . . . . . . 8 |- (x e. V /\ y e. V)
11	10	biantrur 724	. . . . . . 7 |- (z = {f | f:y-->x} <-> ((x e. V /\ y e. V) /\ z = {f | f:y-->x}))
12	11	oprabbii 3992	. . . . . 6 |- {<.<.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 1493	. . . . 5 |- ^m = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | f:y-->x})}
14	4, 6, 13	oprabval2g 4022	. . . 4 |- ((A e. V /\ B e. V /\ {f | f:B-->A} e. V) -> (A ^m B) = {f | f:B-->A})
15	14	3expia 834	. . 3 |- ((A e. V /\ B e. V) -> ({f | f:B-->A} e. V -> (A ^m B) = {f | f:B-->A}))
16		elisset 1814	. . 3 |- (A e. C -> A e. V)
17		elisset 1814	. . 3 |- (B e. D -> B e. V)
18	15, 16, 17	syl2an 454	. 2 |- ((A e. C /\ B e. D) -> ({f | f:B-->A} e. V -> (A ^m B) = {f | f:B-->A}))
19	2, 18	mpd 26	1 |- ((A e. C /\ B e. D) -> (A ^m B) = {f | f:B-->A})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462  Vcvv 1808  -->wf 3174  (class class class)co 3958  {copab2 3959   ^m cm 4315

This theorem is referenced by:  mapval 4325  elmapg 4326  mapsspw 4334  mapss 4339  isfuna 10664

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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