Theorem fconst2g 5643

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 5616	. . . . . . 7 |- ( ( F : A --> { B } /\ x e. A ) -> ( F ` x ) = B )
2	1	adantlr 469	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = B )
3		fvconst2g 5642	. . . . . . 7 |- ( ( B e. C /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
4	3	adantll 468	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
5	2, 4	eqtr4d 2176	. . . . 5 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = ( ( A X. { B } ) ` x ) )
6	5	ralrimiva 2508	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) )
7		ffn 5280	. . . . 5 |- ( F : A --> { B } -> F Fn A )
8		fnconstg 5328	. . . . 5 |- ( B e. C -> ( A X. { B } ) Fn A )
9		eqfnfv 5526	. . . . 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 287	. . . 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 166	. . 3 |- ( ( F : A --> { B } /\ B e. C ) -> F = ( A X. { B } ) )
12	11	expcom 115	. 2 |- ( B e. C -> ( F : A --> { B } -> F = ( A X. { B } ) ) )
13		fconstg 5327	. . 3 |- ( B e. C -> ( A X. { B } ) : A --> { B } )
14		feq1 5263	. . 3 |- ( F = ( A X. { B } ) -> ( F : A --> { B } <-> ( A X. { B } ) : A --> { B } ) )
15	13, 14	syl5ibrcom 156	. 2 |- ( B e. C -> ( F = ( A X. { B } ) -> F : A --> { B } ) )
16	12, 15	impbid 128	1 |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   {csn 3532   X. cxp 4545   Fn wfn 5126   -->wf 5127   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139

This theorem is referenced by:  fconst2  5645  cnconst  12442  nninfall  13379