Theorem fconst2g 5774

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 5747	. . . . . . 7 |- ( ( F : A --> { B } /\ x e. A ) -> ( F ` x ) = B )
2	1	adantlr 477	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = B )
3		fvconst2g 5773	. . . . . . 7 |- ( ( B e. C /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
4	3	adantll 476	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
5	2, 4	eqtr4d 2229	. . . . 5 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = ( ( A X. { B } ) ` x ) )
6	5	ralrimiva 2567	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) )
7		ffn 5404	. . . . 5 |- ( F : A --> { B } -> F Fn A )
8		fnconstg 5452	. . . . 5 |- ( B e. C -> ( A X. { B } ) Fn A )
9		eqfnfv 5656	. . . . 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 289	. . . 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 167	. . 3 |- ( ( F : A --> { B } /\ B e. C ) -> F = ( A X. { B } ) )
12	11	expcom 116	. 2 |- ( B e. C -> ( F : A --> { B } -> F = ( A X. { B } ) ) )
13		fconstg 5451	. . 3 |- ( B e. C -> ( A X. { B } ) : A --> { B } )
14		feq1 5387	. . 3 |- ( F = ( A X. { B } ) -> ( F : A --> { B } <-> ( A X. { B } ) : A --> { B } ) )
15	13, 14	syl5ibrcom 157	. 2 |- ( B e. C -> ( F = ( A X. { B } ) -> F : A --> { B } ) )
16	12, 15	impbid 129	1 |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   {csn 3619   X. cxp 4658   Fn wfn 5250   -->wf 5251   ` cfv 5255

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263

This theorem is referenced by:  fconst2  5776  cnconst  14413  nninfall  15569