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Theorem mptfcl 42736
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 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
Distinct variable groups:   𝑡,𝐴   𝑡,𝐶
Allowed substitution hint:   𝐵(𝑡)

Proof of Theorem mptfcl
StepHypRef Expression
1 eqid 2736 . . 3 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
21fmpt 7129 . 2 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
3 rsp 3246 . 2 (∀𝑡𝐴 𝐵𝐶 → (𝑡𝐴𝐵𝐶))
42, 3sylbir 235 1 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3060  cmpt 5224  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  mzpsubmpt  42759  eq0rabdioph  42792  eqrabdioph  42793
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