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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.
StepHypRef Expression
1 ssrab2 3584 . . 3
2 xpexg 6602 . . . . 5
32ancoms 453 . . . 4
4 pwexg 4636 . . . 4
53, 4syl 16 . . 3
6 ssexg 4598 . . 3
71, 5, 6sylancr 663 . 2
8 elex 3118 . . 3
9 elex 3118 . . 3
10 xpeq2 5019 . . . . . . 7
1110pweqd 4017 . . . . . 6
12 rabeq 3103 . . . . . 6
1311, 12syl 16 . . . . 5
14 xpeq1 5018 . . . . . . 7
1514pweqd 4017 . . . . . 6
16 rabeq 3103 . . . . . 6
1715, 16syl 16 . . . . 5
18 df-pm 7442 . . . . 5
1913, 17, 18ovmpt2g 6437 . . . 4
20193expia 1198 . . 3
218, 9, 20syl2an 477 . 2
227, 21mpd 15 1
 Colors of variables: wff setvar class 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
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