Theorem pmvalg 4331

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 4327	. . 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 3201	. . . . . . . 8 |- (x = A -> (y X. x) = (y X. A))
4	3	sseq2d 2089	. . . . . . 7 |- (x = A -> (f (_ (y X. x) <-> f (_ (y X. A)))
5	4	anbi2d 616	. . . . . 6 |- (x = A -> ((Fun f /\ f (_ (y X. x)) <-> (Fun f /\ f (_ (y X. A))))
6	5	abbidv 1577	. . . . 5 |- (x = A -> {f | (Fun f /\ f (_ (y X. x))} = {f | (Fun f /\ f (_ (y X. A))})
7		xpeq1 3200	. . . . . . . 8 |- (y = B -> (y X. A) = (B X. A))
8	7	sseq2d 2089	. . . . . . 7 |- (y = B -> (f (_ (y X. A) <-> f (_ (B X. A)))
9	8	anbi2d 616	. . . . . 6 |- (y = B -> ((Fun f /\ f (_ (y X. A)) <-> (Fun f /\ f (_ (B X. A))))
10	9	abbidv 1577	. . . . 5 |- (y = B -> {f | (Fun f /\ f (_ (y X. A))} = {f | (Fun f /\ f (_ (B X. A))})
11		df-pm 4325	. . . . . 6 |- ^pm = {<.<.x, y>., z>. | z = {f | (Fun f /\ f (_ (y X. x))}}
12		visset 1813	. . . . . . . . 9 |- x e. V
13		visset 1813	. . . . . . . . 9 |- y e. V
14	12, 13	pm3.2i 285	. . . . . . . 8 |- (x e. V /\ y e. V)
15	14	biantrur 725	. . . . . . 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 3997	. . . . . 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 1495	. . . . 5 |- ^pm = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = {f | (Fun f /\ f (_ (y X. x))})}
18	6, 10, 17	oprabval2g 4027	. . . 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 835	. . 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 1817	. . 3 |- (A e. C -> A e. V)
21		elisset 1817	. . 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 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047   X. cxp 3168  Fun wfun 3176  (class class class)co 3963  {copab2 3964   ^pm cpm 4323

This theorem is referenced by:  elpm 4336

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-pm 4325

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