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Theorem fconst2 6035
Description: A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
Hypothesis
Ref Expression
fvconst2.1  |-  B  e. 
_V
Assertion
Ref Expression
fconst2  |-  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) )

Proof of Theorem fconst2
StepHypRef Expression
1 fvconst2.1 . 2  |-  B  e. 
_V
2 fconst2g 6033 . 2  |-  ( B  e.  _V  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
31, 2ax-mp 5 1  |-  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3070   {csn 3977    X. cxp 4938   -->wf 5514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526
This theorem is referenced by:  map1  7490  rrxcph  21014  dvcmul  21536  plyeq0  21797  lnon0  24335  hsn0elch  24788  df0op2  25293  nmop0h  25532  xrge0mulc1cn  26507  lindsrng01  31111  lfl1  33023  lkr0f  33047
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