Theorem pmresg 7885

Description: Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Assertion

Ref	Expression
pmresg	|- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Proof of Theorem pmresg

Step	Hyp	Ref	Expression
1		n0i 3920	. . . . 5 |- ( F e. ( A ^pm C ) -> -. ( A ^pm C ) = (/) )
2		fnpm 7865	. . . . . . 7 |- ^pm Fn ( _V X. _V )
3		fndm 5990	. . . . . . 7 |- ( ^pm Fn ( _V X. _V ) -> dom ^pm = ( _V X. _V ) )
4	2, 3	ax-mp 5	. . . . . 6 |- dom ^pm = ( _V X. _V )
5	4	ndmov 6818	. . . . 5 |- ( -. ( A e. _V /\ C e. _V ) -> ( A ^pm C ) = (/) )
6	1, 5	nsyl2 142	. . . 4 |- ( F e. ( A ^pm C ) -> ( A e. _V /\ C e. _V ) )
7	6	simpld 475	. . 3 |- ( F e. ( A ^pm C ) -> A e. _V )
8	7	adantl 482	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> A e. _V )
9		simpl 473	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> B e. V )
10		elpmi 7876	. . . . . 6 |- ( F e. ( A ^pm C ) -> ( F : dom F --> A /\ dom F C_ C ) )
11	10	simpld 475	. . . . 5 |- ( F e. ( A ^pm C ) -> F : dom F --> A )
12	11	adantl 482	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> F : dom F --> A )
13		inss1 3833	. . . 4 |- ( dom F i^i B ) C_ dom F
14		fssres 6070	. . . 4 |- ( ( F : dom F --> A /\ ( dom F i^i B ) C_ dom F ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A )
15	12, 13, 14	sylancl 694	. . 3 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A )
16		ffun 6048	. . . . 5 |- ( F : dom F --> A -> Fun F )
17		resres 5409	. . . . . 6 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
18		funrel 5905	. . . . . . 7 |- ( Fun F -> Rel F )
19		resdm 5441	. . . . . . 7 |- ( Rel F -> ( F |` dom F ) = F )
20		reseq1 5390	. . . . . . 7 |- ( ( F |` dom F ) = F -> ( ( F |` dom F ) |` B ) = ( F |` B ) )
21	18, 19, 20	3syl 18	. . . . . 6 |- ( Fun F -> ( ( F |` dom F ) |` B ) = ( F |` B ) )
22	17, 21	syl5eqr 2670	. . . . 5 |- ( Fun F -> ( F |` ( dom F i^i B ) ) = ( F |` B ) )
23	12, 16, 22	3syl 18	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) = ( F |` B ) )
24	23	feq1d 6030	. . 3 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A <-> ( F |` B ) : ( dom F i^i B ) --> A ) )
25	15, 24	mpbid 222	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) : ( dom F i^i B ) --> A )
26		inss2 3834	. . 3 |- ( dom F i^i B ) C_ B
27		elpm2r 7875	. . 3 |- ( ( ( A e. _V /\ B e. V ) /\ ( ( F |` B ) : ( dom F i^i B ) --> A /\ ( dom F i^i B ) C_ B ) ) -> ( F |` B ) e. ( A ^pm B ) )
28	26, 27	mpanr2 720	. 2 |- ( ( ( A e. _V /\ B e. V ) /\ ( F |` B ) : ( dom F i^i B ) --> A ) -> ( F |` B ) e. ( A ^pm B ) )
29	8, 9, 25, 28	syl21anc 1325	1 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   |` cres 5116   Rel wrel 5119   Fun wfun 5882   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:  lmres  21104  mbfres  23411  dvnres  23694  cpnres  23700  caures  33556