Theorem elpm2r 6930

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 5534	. . . . . . 7 |- ( F : C --> A -> dom F = C )
2	1	feq2d 5516	. . . . . 6 |- ( F : C --> A -> ( F : dom F --> A <-> F : C --> A ) )
3	1	sseq1d 3277	. . . . . 6 |- ( F : C --> A -> ( dom F C_ B <-> C C_ B ) )
4	2, 3	anbi12d 477	. . . . 5 |- ( F : C --> A -> ( ( F : dom F --> A /\ dom F C_ B ) <-> ( F : C --> A /\ C C_ B ) ) )
5	4	adantr 276	. . . 4 |- ( ( F : C --> A /\ C C_ B ) -> ( ( F : dom F --> A /\ dom F C_ B ) <-> ( F : C --> A /\ C C_ B ) ) )
6	5	ibir 177	. . 3 |- ( ( F : C --> A /\ C C_ B ) -> ( F : dom F --> A /\ dom F C_ B ) )
7		elpm2g 6929	. . 3 |- ( ( A e. V /\ B e. W ) -> ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) )
8	6, 7	imbitrrid 156	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( F : C --> A /\ C C_ B ) -> F e. ( A ^pm B ) ) )
9	8	imp 124	1 |- ( ( ( A e. V /\ B e. W ) /\ ( F : C --> A /\ C C_ B ) ) -> F e. ( A ^pm B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   C_ wss 3220   dom cdm 4769   -->wf 5368  (class class class)co 6075   ^pm cpm 6913

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pm 6915

This theorem is referenced by:  fpmg  6945  pmresg  6947  ennnfonelemg  13272  lmbrf  15239  ellimc3apf  15684  dvfvalap  15705  dvmulxxbr  15726  dvaddxx  15727  dvmulxx  15728  dviaddf  15729  dvimulf  15730  dvcoapbr  15731  dvmptclx  15742