Theorem elpmi 7876

Description: A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

Assertion

Ref	Expression
elpmi	|- ( F e. ( A ^pm B ) -> ( F : dom F --> A /\ dom F C_ B ) )

Proof of Theorem elpmi

Step	Hyp	Ref	Expression
1		n0i 3920	. . . 4 |- ( F e. ( A ^pm B ) -> -. ( A ^pm B ) = (/) )
2		fnpm 7865	. . . . . 6 |- ^pm Fn ( _V X. _V )
3		fndm 5990	. . . . . 6 |- ( ^pm Fn ( _V X. _V ) -> dom ^pm = ( _V X. _V ) )
4	2, 3	ax-mp 5	. . . . 5 |- dom ^pm = ( _V X. _V )
5	4	ndmov 6818	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> ( A ^pm B ) = (/) )
6	1, 5	nsyl2 142	. . 3 |- ( F e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
7		elpm2g 7874	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) )
8	6, 7	syl 17	. 2 |- ( F e. ( A ^pm B ) -> ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) )
9	8	ibi 256	1 |- ( F e. ( A ^pm B ) -> ( F : dom F --> A /\ dom F C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   Fn wfn 5883   -->wf 5884  (class class class)co 6650   ^pm cpm 7858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pm 7860

This theorem is referenced by:  pmfun  7877  pmresg  7885  equivcau  23098  dvn2bss  23693  mrsubff  31409  mrsubrn  31410  elpmrn  39414  elpmi2  39418  issmflem  40936