Theorem fconst4m 5782

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 5781	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )
2		cnvimass 5032	. . . . . 6 |- ( `' F " { B } ) C_ dom F
3		fndm 5357	. . . . . 6 |- ( F Fn A -> dom F = A )
4	2, 3	sseqtrid 3233	. . . . 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 3198	. . . 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 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   {csn 3622   `'ccnv 4662   dom cdm 4663   "cima 4666   Fn wfn 5253   -->wf 5254

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by: (None)