Theorem pmresg 6844

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 6819	. . . 4 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl1 6217	. . 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 6835	. . . . . 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 3427	. . . 4 |- ( dom F i^i B ) C_ dom F
9		fssres 5512	. . . 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 5485	. . . . 5 |- ( F : dom F --> A -> Fun F )
12		resres 5025	. . . . . 6 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
13		funrel 5343	. . . . . . 7 |- ( Fun F -> Rel F )
14		resdm 5052	. . . . . . 7 |- ( Rel F -> ( F |` dom F ) = F )
15		reseq1 5007	. . . . . . 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 2278	. . . . 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 5469	. . 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 3428	. . 3 |- ( dom F i^i B ) C_ B
22		elpm2r 6834	. . 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 1272	1 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   i^i cin 3199   C_ wss 3200   ~Pcpw 3652   X. cxp 4723   dom cdm 4725   |` cres 4727   Rel wrel 4730   Fun wfun 5320   -->wf 5322  (class class class)co 6017   ^pm cpm 6817

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pm 6819

This theorem is referenced by:  lmres  14971