Theorem pmvalg 4321

Description: The value of the partial mapping operation. (A ^pm B) is the set of all partial functions that map from B to A.

Assertion

Ref	Expression
pmvalg	|- ((A e. C /\ B e. D) -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))})

Distinct variable groups:   A,f   B,f

Proof of Theorem pmvalg

Step	Hyp	Ref	Expression
1		pmex 4317	. . 3 |- ((B e. D /\ A e. C) -> {f | (Fun f /\ f (_ (B X. A))} e. V)
2	1	ancoms 436	. 2 |- ((A e. C /\ B e. D) -> {f | (Fun f /\ f (_ (B X. A))} e. V)
3		xpeq2 3196	. . . . . . . 8 |- (x = A -> (y X. x) = (y X. A))
4	3	sseq2d 2085	. . . . . . 7 |- (x = A -> (f (_ (y X. x) <-> f (_ (y X. A)))
5	4	anbi2d 615	. . . . . 6 |- (x = A -> ((Fun f /\ f (_ (y X. x)) <-> (Fun f /\ f (_ (y X. A))))
6	5	abbidv 1574	. . . . 5 |- (x = A -> {f | (Fun f /\ f (_ (y X. x))} = {f | (Fun f /\ f (_ (y X. A))})
7		xpeq1 3195	. . . . . . . 8 |- (y = B -> (y X. A) = (B X. A))
8	7	sseq2d 2085	. . . . . . 7 |- (y = B -> (f (_ (y X. A) <-> f (_ (B X. A)))
9	8	anbi2d 615	. . . . . 6 |- (y = B -> ((Fun f /\ f (_ (y X. A)) <-> (Fun f /\ f (_ (B X. A))))
10	9	abbidv 1574	. . . . 5 |- (y = B -> {f | (Fun f /\ f (_ (y X. A))} = {f | (Fun f /\ f (_ (B X. A))})
11		df-pm 4315	. . . . . 6 |- ^pm = {<.<.x, y>., z>. | z = {f | (Fun f /\ f (_ (y X. x))}}
12		visset 1809	. . . . . . . . 9 |- x e. V
13		visset 1809	. . . . . . . . 9 |- y e. V
14	12, 13	pm3.2i 285	. . . . . . . 8 |- (x e. V /\ y e. V)
15	14	biantrur 724	. . . . . . 7 |- (z = {f | (Fun f /\ f (_ (y X. x))} <-> ((x e. V /\ y e. V) /\ z = {f | (Fun f /\ f (_ (y X. x))}))
16	15	oprabbii 3988	. . . . . 6 |- {<.<.x, y>., z>. | z = {f | (Fun f /\ f (_ (y X. x))}} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | (Fun f /\ f (_ (y X. x))})}
17	11, 16	eqtr 1492	. . . . 5 |- ^pm = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | (Fun f /\ f (_ (y X. x))})}
18	6, 10, 17	oprabval2g 4018	. . . 4 |- ((A e. V /\ B e. V /\ {f | (Fun f /\ f (_ (B X. A))} e. V) -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))})
19	18	3expia 834	. . 3 |- ((A e. V /\ B e. V) -> ({f | (Fun f /\ f (_ (B X. A))} e. V -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))}))
20		elisset 1813	. . 3 |- (A e. C -> A e. V)
21		elisset 1813	. . 3 |- (B e. D -> B e. V)
22	19, 20, 21	syl2an 454	. 2 |- ((A e. C /\ B e. D) -> ({f | (Fun f /\ f (_ (B X. A))} e. V -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))}))
23	2, 22	mpd 26	1 |- ((A e. C /\ B e. D) -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807   (_ wss 2043   X. cxp 3163  Fun wfun 3171  (class class class)co 3954  {copab2 3955   ^pm cpm 4313

This theorem is referenced by:  elpm 4326

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-rab 1649  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-fv 3193  df-opr 3956  df-oprab 3957  df-pm 4315

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