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Theorem mptfcl 30243
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 2460 . . 3  |-  ( t  e.  A  |->  B )  =  ( t  e.  A  |->  B )
21fmpt 6033 . 2  |-  ( A. t  e.  A  B  e.  C  <->  ( t  e.  A  |->  B ) : A --> C )
3 rsp 2823 . 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 1762   A.wral 2807    |-> cmpt 4498   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  mzpsubmpt  30266  eq0rabdioph  30301  eqrabdioph  30302
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