Theorem elpm2r 7875

Description: Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Assertion

Ref	Expression
elpm2r	|- ( ( ( A e. V /\ B e. W ) /\ ( F : C --> A /\ C C_ B ) ) -> F e. ( A ^pm B ) )

Proof of Theorem elpm2r

Step	Hyp	Ref	Expression
1		fdm 6051	. . . . . . 7 |- ( F : C --> A -> dom F = C )
2	1	feq2d 6031	. . . . . 6 |- ( F : C --> A -> ( F : dom F --> A <-> F : C --> A ) )
3	1	sseq1d 3632	. . . . . 6 |- ( F : C --> A -> ( dom F C_ B <-> C C_ B ) )
4	2, 3	anbi12d 747	. . . . 5 |- ( F : C --> A -> ( ( F : dom F --> A /\ dom F C_ B ) <-> ( F : C --> A /\ C C_ B ) ) )
5	4	adantr 481	. . . 4 |- ( ( F : C --> A /\ C C_ B ) -> ( ( F : dom F --> A /\ dom F C_ B ) <-> ( F : C --> A /\ C C_ B ) ) )
6	5	ibir 257	. . 3 |- ( ( F : C --> A /\ C C_ B ) -> ( F : dom F --> A /\ dom F C_ B ) )
7		elpm2g 7874	. . 3 |- ( ( A e. V /\ B e. W ) -> ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) )
8	6, 7	syl5ibr 236	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( F : C --> A /\ C C_ B ) -> F e. ( A ^pm B ) ) )
9	8	imp 445	1 |- ( ( ( A e. V /\ B e. W ) /\ ( F : C --> A /\ C C_ B ) ) -> F e. ( A ^pm B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   dom cdm 5114   -->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-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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  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-pm 7860

This theorem is referenced by:  fpmg  7883  pmresg  7885  rlim  14226  ello12  14247  elo12  14258  sscpwex  16475  catcfuccl  16759  catcxpccl  16847  lmbrf  21064  cnextfval  21866  lmmbrf  23060  iscauf  23078  caucfil  23081  cmetcaulem  23086  lmclimf  23102  ismbf  23397  ismbfcn  23398  mbfconst  23402  cncombf  23425  cnmbf  23426  limcfval  23636  dvfval  23661  dvnff  23686  dvn2bss  23693  dvnfre  23715  taylfvallem1  24111  taylfval  24113  tayl0  24116  taylplem1  24117  taylply2  24122  taylply  24123  dvtaylp  24124  dvntaylp  24125  dvntaylp0  24126  taylthlem1  24127  taylthlem2  24128  ulmval  24134  ulmpm  24137  iscgrgd  25408  esumcvg  30148  mrsubfval  31405  elmrsubrn  31417  msubfval  31421  fwddifval  32269  fwddifnval  32270  fpmd  39483  xlimmnfvlem2  40059  xlimpnfvlem2  40063  dvnmptdivc  40153  dvnxpaek  40157  etransclem46  40497  issmflem  40936  fdivpm  42337  refdivpm  42338  elbigo2  42346