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Theorem mptfcl 41090
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 2733 . . 3 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
21fmpt 7062 . 2 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
3 rsp 3229 . 2 (∀𝑡𝐴 𝐵𝐶 → (𝑡𝐴𝐵𝐶))
42, 3sylbir 234 1 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3061  cmpt 5192  wf 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-fun 6502  df-fn 6503  df-f 6504
This theorem is referenced by:  mzpsubmpt  41113  eq0rabdioph  41146  eqrabdioph  41147
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