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Theorem mptfcl 37802
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 2770 . . 3 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
21fmpt 6523 . 2 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
3 rsp 3077 . 2 (∀𝑡𝐴 𝐵𝐶 → (𝑡𝐴𝐵𝐶))
42, 3sylbir 225 1 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  wral 3060  cmpt 4861  wf 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  mzpsubmpt  37825  eq0rabdioph  37859  eqrabdioph  37860
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