MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  strlemor0 Structured version   Visualization version   GIF version

Theorem strlemor0 15884
Description: Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Assertion
Ref Expression
strlemor0 (Fun ∅ ∧ dom ∅ ⊆ (1...0))

Proof of Theorem strlemor0
StepHypRef Expression
1 fun0 5914 . . 3 Fun ∅
2 funcnvcnv 5916 . . 3 (Fun ∅ → Fun ∅)
31, 2ax-mp 5 . 2 Fun
4 dm0 5303 . . 3 dom ∅ = ∅
5 0ss 3949 . . 3 ∅ ⊆ (1...0)
64, 5eqsstri 3619 . 2 dom ∅ ⊆ (1...0)
73, 6pm3.2i 471 1 (Fun ∅ ∧ dom ∅ ⊆ (1...0))
Colors of variables: wff setvar class
Syntax hints:  wa 384  wss 3560  c0 3896  ccnv 5078  dom cdm 5079  Fun wfun 5844  (class class class)co 6605  0cc0 9881  1c1 9882  ...cfz 12265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-fun 5852
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator