Theorem fconst2g 5376

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 5351	. . . . . . 7 |- ( ( F : A --> { B } /\ x e. A ) -> ( F ` x ) = B )
2	1	adantlr 446	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = B )
3		fvconst2g 5375	. . . . . . 7 |- ( ( B e. C /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
4	3	adantll 445	. . . . . 6 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( ( A X. { B } ) ` x ) = B )
5	2, 4	eqtr4d 2075	. . . . 5 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = ( ( A X. { B } ) ` x ) )
6	5	ralrimiva 2392	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) )
7		ffn 5046	. . . . 5 |- ( F : A --> { B } -> F Fn A )
8		fnconstg 5084	. . . . 5 |- ( B e. C -> ( A X. { B } ) Fn A )
9		eqfnfv 5265	. . . . 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 273	. . . 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 156	. . 3 |- ( ( F : A --> { B } /\ B e. C ) -> F = ( A X. { B } ) )
12	11	expcom 109	. 2 |- ( B e. C -> ( F : A --> { B } -> F = ( A X. { B } ) ) )
13		fconstg 5083	. . 3 |- ( B e. C -> ( A X. { B } ) : A --> { B } )
14		feq1 5030	. . 3 |- ( F = ( A X. { B } ) -> ( F : A --> { B } <-> ( A X. { B } ) : A --> { B } ) )
15	13, 14	syl5ibrcom 146	. 2 |- ( B e. C -> ( F = ( A X. { B } ) -> F : A --> { B } ) )
16	12, 15	impbid 120	1 |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   {csn 3375   X. cxp 4343   Fn wfn 4897   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by:  fconst2  5378