Theorem pmvalg 5390

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 C_ (B X. A))})

Distinct variable groups:   A,f   B,f

Proof of Theorem pmvalg

Step	Hyp	Ref	Expression
1		pmex 5386	. . 3 |- ((B e. D /\ A e. C) -> {f | (Fun f /\ f C_ (B X. A))} e. _V)
2	1	ancoms 484	. 2 |- ((A e. C /\ B e. D) -> {f | (Fun f /\ f C_ (B X. A))} e. _V)
3		xpeq2 4017	. . . . . . . 8 |- (x = A -> (y X. x) = (y X. A))
4	3	sseq2d 2645	. . . . . . 7 |- (x = A -> (f C_ (y X. x) <-> f C_ (y X. A)))
5	4	anbi2d 678	. . . . . 6 |- (x = A -> ((Fun f /\ f C_ (y X. x)) <-> (Fun f /\ f C_ (y X. A))))
6	5	abbidv 2008	. . . . 5 |- (x = A -> {f | (Fun f /\ f C_ (y X. x))} = {f | (Fun f /\ f C_ (y X. A))})
7		xpeq1 4016	. . . . . . . 8 |- (y = B -> (y X. A) = (B X. A))
8	7	sseq2d 2645	. . . . . . 7 |- (y = B -> (f C_ (y X. A) <-> f C_ (B X. A)))
9	8	anbi2d 678	. . . . . 6 |- (y = B -> ((Fun f /\ f C_ (y X. A)) <-> (Fun f /\ f C_ (B X. A))))
10	9	abbidv 2008	. . . . 5 |- (y = B -> {f | (Fun f /\ f C_ (y X. A))} = {f | (Fun f /\ f C_ (B X. A))})
11		df-pm 5384	. . . . . 6 |- ^pm = {<.<.x, y>., z>. | z = {f | (Fun f /\ f C_ (y X. x))}}
12		visset 2295	. . . . . . . . 9 |- x e. _V
13		visset 2295	. . . . . . . . 9 |- y e. _V
14	12, 13	pm3.2i 307	. . . . . . . 8 |- (x e. _V /\ y e. _V)
15	14	biantrur 794	. . . . . . 7 |- (z = {f | (Fun f /\ f C_ (y X. x))} <-> ((x e. _V /\ y e. _V) /\ z = {f | (Fun f /\ f C_ (y X. x))}))
16	15	oprabbii 4923	. . . . . 6 |- {<.<.x, y>., z>. | z = {f | (Fun f /\ f C_ (y X. x))}} = {<.<.x, y>., z>. | ((x e. _V /\ y e. _V) /\ z = {f | (Fun f /\ f C_ (y X. x))})}
17	11, 16	eqtri 1908	. . . . 5 |- ^pm = {<.<.x, y>., z>. | ((x e. _V /\ y e. _V) /\ z = {f | (Fun f /\ f C_ (y X. x))})}
18	6, 10, 17	oprabval2g 4956	. . . 4 |- ((A e. _V /\ B e. _V /\ {f | (Fun f /\ f C_ (B X. A))} e. _V) -> (A ^pm B) = {f | (Fun f /\ f C_ (B X. A))})
19	18	3expia 1069	. . 3 |- ((A e. _V /\ B e. _V) -> ({f | (Fun f /\ f C_ (B X. A))} e. _V -> (A ^pm B) = {f | (Fun f /\ f C_ (B X. A))}))
20		elisset 2299	. . 3 |- (A e. C -> A e. _V)
21		elisset 2299	. . 3 |- (B e. D -> B e. _V)
22	19, 20, 21	syl2an 503	. 2 |- ((A e. C /\ B e. D) -> ({f | (Fun f /\ f C_ (B X. A))} e. _V -> (A ^pm B) = {f | (Fun f /\ f C_ (B X. A))}))
23	2, 22	mpd 29	1 |- ((A e. C /\ B e. D) -> (A ^pm B) = {f | (Fun f /\ f C_ (B X. A))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593   X. cxp 3984  Fun wfun 3992  (class class class)co 4884  {copab2 4885   ^pm cpm 5382

This theorem is referenced by:  elpm 5395

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-pm 5384

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