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Theorem mptfcl 40137
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 2739 . . 3 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
21fmpt 6887 . 2 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
3 rsp 3119 . 2 (∀𝑡𝐴 𝐵𝐶 → (𝑡𝐴𝐵𝐶))
42, 3sylbir 238 1 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wral 3054  cmpt 5111  wf 6336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-fun 6342  df-fn 6343  df-f 6344
This theorem is referenced by:  mzpsubmpt  40160  eq0rabdioph  40193  eqrabdioph  40194
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