Theorem fconst4m 5926

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 5925	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )
2		cnvimass 5145	. . . . . 6 |- ( `' F " { B } ) C_ dom F
3		fndm 5475	. . . . . 6 |- ( F Fn A -> dom F = A )
4	2, 3	sseqtrid 3298	. . . . 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 3263	. . . 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 458	. 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 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   {csn 3705   `'ccnv 4768   dom cdm 4769   "cima 4772   Fn wfn 5367   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380

This theorem is referenced by: (None)