Theorem pmresg 6762

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

Dummy variables x f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pm 6737	. . . 4 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl1 6141	. . 3 |- ( F e. ( A ^pm C ) -> A e. _V )
3	2	adantl 277	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> A e. _V )
4		simpl 109	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> B e. V )
5		elpmi 6753	. . . . . 6 |- ( F e. ( A ^pm C ) -> ( F : dom F --> A /\ dom F C_ C ) )
6	5	simpld 112	. . . . 5 |- ( F e. ( A ^pm C ) -> F : dom F --> A )
7	6	adantl 277	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> F : dom F --> A )
8		inss1 3392	. . . 4 |- ( dom F i^i B ) C_ dom F
9		fssres 5450	. . . 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 )
10	7, 8, 9	sylancl 413	. . 3 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A )
11		ffun 5427	. . . . 5 |- ( F : dom F --> A -> Fun F )
12		resres 4970	. . . . . 6 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
13		funrel 5287	. . . . . . 7 |- ( Fun F -> Rel F )
14		resdm 4997	. . . . . . 7 |- ( Rel F -> ( F |` dom F ) = F )
15		reseq1 4952	. . . . . . 7 |- ( ( F |` dom F ) = F -> ( ( F |` dom F ) |` B ) = ( F |` B ) )
16	13, 14, 15	3syl 17	. . . . . 6 |- ( Fun F -> ( ( F |` dom F ) |` B ) = ( F |` B ) )
17	12, 16	eqtr3id 2251	. . . . 5 |- ( Fun F -> ( F |` ( dom F i^i B ) ) = ( F |` B ) )
18	7, 11, 17	3syl 17	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) = ( F |` B ) )
19	18	feq1d 5411	. . 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 ) )
20	10, 19	mpbid 147	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) : ( dom F i^i B ) --> A )
21		inss2 3393	. . 3 |- ( dom F i^i B ) C_ B
22		elpm2r 6752	. . 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 ) )
23	21, 22	mpanr2 438	. 2 |- ( ( ( A e. _V /\ B e. V ) /\ ( F |` B ) : ( dom F i^i B ) --> A ) -> ( F |` B ) e. ( A ^pm B ) )
24	3, 4, 20, 23	syl21anc 1248	1 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {crab 2487   _Vcvv 2771   i^i cin 3164   C_ wss 3165   ~Pcpw 3615   X. cxp 4672   dom cdm 4674   |` cres 4676   Rel wrel 4679   Fun wfun 5264   -->wf 5266  (class class class)co 5943   ^pm cpm 6735

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pm 6737

This theorem is referenced by:  lmres  14691