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Theorem strlemor0 13250
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 5323 . . 3  |-  Fun  (/)
2 funcnvcnv 5324 . . 3  |-  ( Fun  (/)  ->  Fun  `' `' (/) )
31, 2ax-mp 8 . 2  |-  Fun  `' `' (/)
4 dm0 4908 . . 3  |-  dom  (/)  =  (/)
5 0ss 3496 . . 3  |-  (/)  C_  (
1 ... 0 )
64, 5eqsstri 3221 . 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 3165   (/)c0 3468   `'ccnv 4704   dom cdm 4705   Fun wfun 5265  (class class class)co 5874   0cc0 8753   1c1 8754   ...cfz 10798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273
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