Theorem fnpm 7506

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 7501	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 3060	. . . . 5 |- y e. _V
3		vex 3060	. . . . 5 |- x e. _V
4	2, 3	xpex 6622	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4600	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4568	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpt2i 6889	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2753   _Vcvv 3057   ~Pcpw 3963   X. cxp 4851   Fun wfun 5595   Fn wfn 5596   ^pm cpm 7499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-pm 7501

This theorem is referenced by:  elpmi  7516  pmresg  7525  pmsspw  7532