Theorem mapvalg 4320

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 4318	. . 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 3614	. . . . . 6 |- (x = A -> (f:y-->x <-> f:y-->A))
4	3	abbidv 1574	. . . . 5 |- (x = A -> {f | f:y-->x} = {f | f:y-->A})
5		feq2 3613	. . . . . 6 |- (y = B -> (f:y-->A <-> f:B-->A))
6	5	abbidv 1574	. . . . 5 |- (y = B -> {f | f:y-->A} = {f | f:B-->A})
7		df-map 4314	. . . . . 6 |- ^m = {<.<.x, y>., z>. | z = {f | f:y-->x}}
8		visset 1809	. . . . . . . . 9 |- x e. V
9		visset 1809	. . . . . . . . 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 3988	. . . . . 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 1492	. . . . 5 |- ^m = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | f:y-->x})}
14	4, 6, 13	oprabval2g 4018	. . . 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 1813	. . 3 |- (A e. C -> A e. V)
17		elisset 1813	. . 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 954   e. wcel 956  {cab 1461  Vcvv 1807  -->wf 3173  (class class class)co 3954  {copab2 3955   ^m cm 4312

This theorem is referenced by:  mapval 4322  elmapg 4323  mapsspw 4331  mapss 4336  isfuna 10628

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-oprab 3957  df-map 4314

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