Theorem fconst2g 5397

Description: A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Assertion

Ref	Expression
fconst2g	|- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )

Proof of Theorem fconst2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvconst 5372	. . . . . . 7 |- ( ( F : A --> { B } /\ x e. A ) -> ( F ` x ) = B )
2	1	adantlr 460	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = B )
3		fvconst2g 5396	. . . . . . 7 |- ( ( B e. C /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
4	3	adantll 459	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
5	2, 4	eqtr4d 2116	. . . . 5 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = ( ( A X. { B } ) ` x ) )
6	5	ralrimiva 2434	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) )
7		ffn 5066	. . . . 5 |- ( F : A --> { B } -> F Fn A )
8		fnconstg 5104	. . . . 5 |- ( B e. C -> ( A X. { B } ) Fn A )
9		eqfnfv 5286	. . . . 5 |- ( ( F Fn A /\ ( A X. { B } ) Fn A ) -> ( F = ( A X. { B } ) <-> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) ) )
10	7, 8, 9	syl2an 283	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> ( F = ( A X. { B } ) <-> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) ) )
11	6, 10	mpbird 165	. . 3 |- ( ( F : A --> { B } /\ B e. C ) -> F = ( A X. { B } ) )
12	11	expcom 114	. 2 |- ( B e. C -> ( F : A --> { B } -> F = ( A X. { B } ) ) )
13		fconstg 5103	. . 3 |- ( B e. C -> ( A X. { B } ) : A --> { B } )
14		feq1 5050	. . 3 |- ( F = ( A X. { B } ) -> ( F : A --> { B } <-> ( A X. { B } ) : A --> { B } ) )
15	13, 14	syl5ibrcom 155	. 2 |- ( B e. C -> ( F = ( A X. { B } ) -> F : A --> { B } ) )
16	12, 15	impbid 127	1 |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   {csn 3398   X. cxp 4361   Fn wfn 4917   -->wf 4918   ` cfv 4922

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by:  fconst2  5399