Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mptfcl Structured version   Visualization version   GIF version

Theorem mptfcl 38709
 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 2779 . . 3 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
21fmpt 6697 . 2 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
3 rsp 3156 . 2 (∀𝑡𝐴 𝐵𝐶 → (𝑡𝐴𝐵𝐶))
42, 3sylbir 227 1 ((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2050  ∀wral 3089   ↦ cmpt 5008  ⟶wf 6184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196 This theorem is referenced by:  mzpsubmpt  38732  eq0rabdioph  38766  eqrabdioph  38767
 Copyright terms: Public domain W3C validator