Theorem pmvalg 7450

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

Assertion

Ref	Expression
pmvalg

Distinct variable groups:   ,   ,

Proof of Theorem pmvalg

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

Step	Hyp	Ref	Expression
1		ssrab2 3584	. . 3
2		xpexg 6602	. . . . 5
3	2	ancoms 453	. . . 4
4		pwexg 4636	. . . 4
5	3, 4	syl 16	. . 3
6		ssexg 4598	. . 3
7	1, 5, 6	sylancr 663	. 2
8		elex 3118	. . 3
9		elex 3118	. . 3
10		xpeq2 5019	. . . . . . 7
11	10	pweqd 4017	. . . . . 6
12		rabeq 3103	. . . . . 6
13	11, 12	syl 16	. . . . 5
14		xpeq1 5018	. . . . . . 7
15	14	pweqd 4017	. . . . . 6
16		rabeq 3103	. . . . . 6
17	15, 16	syl 16	. . . . 5
18		df-pm 7442	. . . . 5
19	13, 17, 18	ovmpt2g 6437	. . . 4
20	19	3expia 1198	. . 3
21	8, 9, 20	syl2an 477	. 2
22	7, 21	mpd 15	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {crab 2811  cvv 3109  C_wss 3475  ~Pcpw 4012  X.cxp 5002  Funwfun 5587  (class class class)co 6296  cpm 7440

This theorem is referenced by:  elpmg  7454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-pm 7442