Theorem fconst2g 5700

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 5673	. . . . . . 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 5699	. . . . . . 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 2201	. . . . 5 |- ( ( ( F : A --> { B } /\ B e. C ) /\ x e. A ) -> ( F ` x ) = ( ( A X. { B } ) ` x ) )
6	5	ralrimiva 2539	. . . 4 |- ( ( F : A --> { B } /\ B e. C ) -> A. x e. A ( F ` x ) = ( ( A X. { B } ) ` x ) )
7		ffn 5337	. . . . 5 |- ( F : A --> { B } -> F Fn A )
8		fnconstg 5385	. . . . 5 |- ( B e. C -> ( A X. { B } ) Fn A )
9		eqfnfv 5583	. . . . 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 5384	. . 3 |- ( B e. C -> ( A X. { B } ) : A --> { B } )
14		feq1 5320	. . 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 1343   e. wcel 2136   A.wral 2444   {csn 3576   X. cxp 4602   Fn wfn 5183   -->wf 5184   ` cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by:  fconst2  5702  cnconst  12874  nninfall  13889