Theorem fconst4m 5749

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 5748	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )
2		cnvimass 5003	. . . . . 6 |- ( `' F " { B } ) C_ dom F
3		fndm 5327	. . . . . 6 |- ( F Fn A -> dom F = A )
4	2, 3	sseqtrid 3217	. . . . 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 3182	. . . 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 1363   E.wex 1502   e. wcel 2158   C_ wss 3141   {csn 3604   `'ccnv 4637   dom cdm 4638   "cima 4641   Fn wfn 5223   -->wf 5224

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236

This theorem is referenced by: (None)