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Theorem strlemor0 13234
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  (/)  C_  (
1 ... 0 ) )

Proof of Theorem strlemor0
StepHypRef Expression
1 fun0 5307 . . 3  |-  Fun  (/)
2 funcnvcnv 5308 . . 3  |-  ( Fun  (/)  ->  Fun  `' `' (/) )
31, 2ax-mp 8 . 2  |-  Fun  `' `' (/)
4 dm0 4892 . . 3  |-  dom  (/)  =  (/)
5 0ss 3483 . . 3  |-  (/)  C_  (
1 ... 0 )
64, 5eqsstri 3208 . 2  |-  dom  (/)  C_  (
1 ... 0 )
73, 6pm3.2i 441 1  |-  ( Fun  `' `' (/)  /\  dom  (/)  C_  (
1 ... 0 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    C_ wss 3152   (/)c0 3455   `'ccnv 4688   dom cdm 4689   Fun wfun 5249  (class class class)co 5858   0cc0 8737   1c1 8738   ...cfz 10782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257
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