Theorem fnpm 6824

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 6819	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 2805	. . . . 5 |- y e. _V
3		vex 2805	. . . . 5 |- x e. _V
4	2, 3	xpex 4842	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4273	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4234	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpoi 6367	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2514   _Vcvv 2802   ~Pcpw 3652   X. cxp 4723   Fun wfun 5320   Fn wfn 5321   ^pm cpm 6817

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pm 6819

This theorem is referenced by:  lmfval  14916