Theorem pmresg 6387

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 6362	. . . 4 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpt2cl1 5802	. . 3 |- ( F e. ( A ^pm C ) -> A e. _V )
3	2	adantl 271	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> A e. _V )
4		simpl 107	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> B e. V )
5		elpmi 6378	. . . . . 6 |- ( F e. ( A ^pm C ) -> ( F : dom F --> A /\ dom F C_ C ) )
6	5	simpld 110	. . . . 5 |- ( F e. ( A ^pm C ) -> F : dom F --> A )
7	6	adantl 271	. . . 4 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> F : dom F --> A )
8		inss1 3209	. . . 4 |- ( dom F i^i B ) C_ dom F
9		fssres 5152	. . . 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 404	. . 3 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> A )
11		ffun 5131	. . . . 5 |- ( F : dom F --> A -> Fun F )
12		resres 4695	. . . . . 6 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
13		funrel 5000	. . . . . . 7 |- ( Fun F -> Rel F )
14		resdm 4720	. . . . . . 7 |- ( Rel F -> ( F |` dom F ) = F )
15		reseq1 4677	. . . . . . 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 2131	. . . . 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 5116	. . 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 145	. 2 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) : ( dom F i^i B ) --> A )
21		inss2 3210	. . 3 |- ( dom F i^i B ) C_ B
22		elpm2r 6377	. . 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 429	. 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 1171	1 |- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1287   e. wcel 1436   {crab 2359   _Vcvv 2615   i^i cin 2987   C_ wss 2988   ~Pcpw 3415   X. cxp 4411   dom cdm 4413   |` cres 4415   Rel wrel 4418   Fun wfun 4977   -->wf 4979  (class class class)co 5615   ^pm cpm 6360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pm 6362

This theorem is referenced by: (None)