Theorem pmvalg 6637

Description: The value of the partial mapping operation. ( A ^pm B ) is the set of all partial functions that map from B to A. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   C( f)   D( f)

Proof of Theorem pmvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3232	. . 3 |- { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A )
2		xpexg 4725	. . . . 5 |- ( ( B e. D /\ A e. C ) -> ( B X. A ) e. _V )
3	2	ancoms 266	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( B X. A ) e. _V )
4		pwexg 4166	. . . 4 |- ( ( B X. A ) e. _V -> ~P ( B X. A ) e. _V )
5	3, 4	syl 14	. . 3 |- ( ( A e. C /\ B e. D ) -> ~P ( B X. A ) e. _V )
6		ssexg 4128	. . 3 |- ( ( { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A ) /\ ~P ( B X. A ) e. _V ) -> { f e. ~P ( B X. A ) | Fun f } e. _V )
7	1, 5, 6	sylancr 412	. 2 |- ( ( A e. C /\ B e. D ) -> { f e. ~P ( B X. A ) | Fun f } e. _V )
8		elex 2741	. . 3 |- ( A e. C -> A e. _V )
9		elex 2741	. . 3 |- ( B e. D -> B e. _V )
10		xpeq2 4626	. . . . . . 7 |- ( x = A -> ( y X. x ) = ( y X. A ) )
11	10	pweqd 3571	. . . . . 6 |- ( x = A -> ~P ( y X. x ) = ~P ( y X. A ) )
12		rabeq 2722	. . . . . 6 |- ( ~P ( y X. x ) = ~P ( y X. A ) -> { f e. ~P ( y X. x ) | Fun f } = { f e. ~P ( y X. A ) | Fun f } )
13	11, 12	syl 14	. . . . 5 |- ( x = A -> { f e. ~P ( y X. x ) | Fun f } = { f e. ~P ( y X. A ) | Fun f } )
14		xpeq1 4625	. . . . . . 7 |- ( y = B -> ( y X. A ) = ( B X. A ) )
15	14	pweqd 3571	. . . . . 6 |- ( y = B -> ~P ( y X. A ) = ~P ( B X. A ) )
16		rabeq 2722	. . . . . 6 |- ( ~P ( y X. A ) = ~P ( B X. A ) -> { f e. ~P ( y X. A ) | Fun f } = { f e. ~P ( B X. A ) | Fun f } )
17	15, 16	syl 14	. . . . 5 |- ( y = B -> { f e. ~P ( y X. A ) | Fun f } = { f e. ~P ( B X. A ) | Fun f } )
18		df-pm 6629	. . . . 5 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
19	13, 17, 18	ovmpog 5987	. . . 4 |- ( ( A e. _V /\ B e. _V /\ { f e. ~P ( B X. A ) | Fun f } e. _V ) -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } )
20	19	3expia 1200	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( { f e. ~P ( B X. A ) | Fun f } e. _V -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } ) )
21	8, 9, 20	syl2an 287	. 2 |- ( ( A e. C /\ B e. D ) -> ( { f e. ~P ( B X. A ) | Fun f } e. _V -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } ) )
22	7, 21	mpd 13	1 |- ( ( A e. C /\ B e. D ) -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   X. cxp 4609   Fun wfun 5192  (class class class)co 5853   ^pm cpm 6627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pm 6629

This theorem is referenced by:  elpmg  6642