Theorem fnpm 7488

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 7483	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 3090	. . . . 5 |- y e. _V
3		vex 3090	. . . . 5 |- x e. _V
4	2, 3	xpex 6609	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4608	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4576	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpt2i 6876	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2786   _Vcvv 3087   ~Pcpw 3985   X. cxp 4852   Fun wfun 5595   Fn wfn 5596   ^pm cpm 7481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-pm 7483

This theorem is referenced by:  elpmi  7498  pmresg  7507  pmsspw  7514