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Theorem mptfcl 30892
Description: Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
mptfcl  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Distinct variable groups:    t, A    t, C
Allowed substitution hint:    B( t)

Proof of Theorem mptfcl
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( t  e.  A  |->  B )  =  ( t  e.  A  |->  B )
21fmpt 6028 . 2  |-  ( A. t  e.  A  B  e.  C  <->  ( t  e.  A  |->  B ) : A --> C )
3 rsp 2820 . 2  |-  ( A. t  e.  A  B  e.  C  ->  ( t  e.  A  ->  B  e.  C ) )
42, 3sylbir 213 1  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   A.wral 2804    |-> cmpt 4497   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  mzpsubmpt  30915  eq0rabdioph  30949  eqrabdioph  30950
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