Theorem pmresg 6578

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 6553	. . . 4 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl1 5977	. . 3 |- ( F e. ( A ^pm C ) -> A e. _V )
3	2	adantl 275	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> A e. _V )
4		simpl 108	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> B e. V )
5		elpmi 6569	. . . . . 6 |- ( F e. ( A ^pm C ) -> ( F : dom F --> A /\ dom F C_ C ) )
6	5	simpld 111	. . . . 5 |- ( F e. ( A ^pm C ) -> F : dom F --> A )
7	6	adantl 275	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> F : dom F --> A )
8		inss1 3301	. . . 4 |- ( dom F i^i B ) C_ dom F
9		fssres 5306	. . . 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 410	. . 3 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A )
11		ffun 5283	. . . . 5 |- ( F : dom F --> A -> Fun F )
12		resres 4839	. . . . . 6 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
13		funrel 5148	. . . . . . 7 |- ( Fun F -> Rel F )
14		resdm 4866	. . . . . . 7 |- ( Rel F -> ( F |` dom F ) = F )
15		reseq1 4821	. . . . . . 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	syl5eqr 2187	. . . . 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 5267	. . 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 146	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) : ( dom F i^i B ) --> A )
21		inss2 3302	. . 3 |- ( dom F i^i B ) C_ B
22		elpm2r 6568	. . 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 435	. 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 1216	1 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {crab 2421   _Vcvv 2689   i^i cin 3075   C_ wss 3076   ~Pcpw 3515   X. cxp 4545   dom cdm 4547   |` cres 4549   Rel wrel 4552   Fun wfun 5125   -->wf 5127  (class class class)co 5782   ^pm cpm 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pm 6553

This theorem is referenced by:  lmres  12456