Theorem fnpm 6985

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 6980	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 2919	. . . . 5 |- y e. _V
3		vex 2919	. . . . 5 |- x e. _V
4	2, 3	xpex 4949	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4342	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4314	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpt2i 6379	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2670   _Vcvv 2916   ~Pcpw 3759   X. cxp 4835   Fun wfun 5407   Fn wfn 5408   ^pm cpm 6978

This theorem is referenced by:  elpmi  6994  pmresg  7000  pmsspw  7007

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-pm 6980