Theorem fnpm 6801

Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
fnpm	|- ^pm Fn ( _V X. _V )

Proof of Theorem fnpm

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

Step	Hyp	Ref	Expression
1		df-pm 6796	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 2802	. . . . 5 |- y e. _V
3		vex 2802	. . . . 5 |- x e. _V
4	2, 3	xpex 4833	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4266	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4227	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpoi 6347	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2512   _Vcvv 2799   ~Pcpw 3649   X. cxp 4716   Fun wfun 5311   Fn wfn 5312   ^pm cpm 6794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pm 6796

This theorem is referenced by:  lmfval  14860