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Theorem strlemor0 13555
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 5508 . . 3  |-  Fun  (/)
2 funcnvcnv 5509 . . 3  |-  ( Fun  (/)  ->  Fun  `' `' (/) )
31, 2ax-mp 8 . 2  |-  Fun  `' `' (/)
4 dm0 5083 . . 3  |-  dom  (/)  =  (/)
5 0ss 3656 . . 3  |-  (/)  C_  (
1 ... 0 )
64, 5eqsstri 3378 . 2  |-  dom  (/)  C_  (
1 ... 0 )
73, 6pm3.2i 442 1  |-  ( Fun  `' `' (/)  /\  dom  (/)  C_  (
1 ... 0 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    C_ wss 3320   (/)c0 3628   `'ccnv 4877   dom cdm 4878   Fun wfun 5448  (class class class)co 6081   0cc0 8990   1c1 8991   ...cfz 11043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456
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