Theorem fconst4m 5882

Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Assertion

Ref	Expression
fconst4m	|- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ ( `' F " { B } ) = A ) ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fconst4m

Step	Hyp	Ref	Expression
1		fconst3m 5881	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )
2		cnvimass 5106	. . . . . 6 |- ( `' F " { B } ) C_ dom F
3		fndm 5436	. . . . . 6 |- ( F Fn A -> dom F = A )
4	2, 3	sseqtrid 3278	. . . . 5 |- ( F Fn A -> ( `' F " { B } ) C_ A )
5	4	biantrurd 305	. . . 4 |- ( F Fn A -> ( A C_ ( `' F " { B } ) <-> ( ( `' F " { B } ) C_ A /\ A C_ ( `' F " { B } ) ) ) )
6		eqss 3243	. . . 4 |- ( ( `' F " { B } ) = A <-> ( ( `' F " { B } ) C_ A /\ A C_ ( `' F " { B } ) ) )
7	5, 6	bitr4di 198	. . 3 |- ( F Fn A -> ( A C_ ( `' F " { B } ) <-> ( `' F " { B } ) = A ) )
8	7	pm5.32i 454	. 2 |- ( ( F Fn A /\ A C_ ( `' F " { B } ) ) <-> ( F Fn A /\ ( `' F " { B } ) = A ) )
9	1, 8	bitrdi 196	1 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ ( `' F " { B } ) = A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   C_ wss 3201   {csn 3673   `'ccnv 4730   dom cdm 4731   "cima 4734   Fn wfn 5328   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341

This theorem is referenced by: (None)