Theorem fnpm 6009

Description: Partial function exponentiation has a universal domain. (Contributed by set.mm contributors, 14-Nov-2013.) (Revised by Scott Fenton, 19-Apr-2019.)

Assertion

Ref	Expression
fnpm	|- ^pm Fn _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 6003	. . 3 |- ^pm = (x e. _V, y e. _V |-> {f e. ~P(y X. x) | Fun f})
2		elin 3220	. . . . . 6 |- (f e. (~P(y X. x) i^i Funs ) <-> (f e. ~P(y X. x) /\ f e. Funs ))
3	2	abbi2i 2465	. . . . 5 |- (~P(y X. x) i^i Funs ) = {f | (f e. ~P(y X. x) /\ f e. Funs )}
4		df-rab 2624	. . . . 5 |- {f e. ~P(y X. x) | f e. Funs } = {f | (f e. ~P(y X. x) /\ f e. Funs )}
5		vex 2863	. . . . . . . 8 |- f e. _V
6	5	elfuns 5830	. . . . . . 7 |- (f e. Funs <-> Fun f)
7	6	rgenw 2682	. . . . . 6 |- A.f e. ~P (y X. x)(f e. Funs <-> Fun f)
8		rabbi 2790	. . . . . 6 |- (A.f e. ~P (y X. x)(f e. Funs <-> Fun f) <-> {f e. ~P(y X. x) | f e. Funs } = {f e. ~P(y X. x) | Fun f})
9	7, 8	mpbi 199	. . . . 5 |- {f e. ~P(y X. x) | f e. Funs } = {f e. ~P(y X. x) | Fun f}
10	3, 4, 9	3eqtr2i 2379	. . . 4 |- (~P(y X. x) i^i Funs ) = {f e. ~P(y X. x) | Fun f}
11		vex 2863	. . . . . . 7 |- y e. _V
12		vex 2863	. . . . . . 7 |- x e. _V
13	11, 12	xpex 5116	. . . . . 6 |- (y X. x) e. _V
14	13	pwex 4330	. . . . 5 |- ~P(y X. x) e. _V
15		funsex 5829	. . . . 5 |- Funs e. _V
16	14, 15	inex 4106	. . . 4 |- (~P(y X. x) i^i Funs ) e. _V
17	10, 16	eqeltrri 2424	. . 3 |- {f e. ~P(y X. x) | Fun f} e. _V
18	1, 17	fnmpt2i 5734	. 2 |- ^pm Fn (_V X. _V)
19		xpvv 4844	. . 3 |- (_V X. _V) = _V
20	19	fneq2i 5180	. 2 |- ( ^pm Fn (_V X. _V) <-> ^pm Fn _V)
21	18, 20	mpbi 199	1 |- ^pm Fn _V

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619  _Vcvv 2860   i^i cin 3209  ~Pcpw 3723   X. cxp 4771  Fun wfun 4776   Fn wfn 4777   Funs cfuns 5760   ^pm cpm 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-funs 5761  df-pm 6003

This theorem is referenced by: (None)