Theorem fnpm 6868

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 6863	. 2 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2		vex 2806	. . . . 5 |- y e. _V
3		vex 2806	. . . . 5 |- x e. _V
4	2, 3	xpex 4848	. . . 4 |- ( y X. x ) e. _V
5	4	pwex 4279	. . 3 |- ~P ( y X. x ) e. _V
6	5	rabex 4239	. 2 |- { f e. ~P ( y X. x ) | Fun f } e. _V
7	1, 6	fnmpoi 6377	1 |- ^pm Fn ( _V X. _V )

Syntax hints:   {crab 2515   _Vcvv 2803   ~Pcpw 3656   X. cxp 4729   Fun wfun 5327   Fn wfn 5328   ^pm cpm 6861

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pm 6863

This theorem is referenced by:  lmfval  14987