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Theorem wemappo 7995
 Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wemapso.t
Assertion
Ref Expression
wemappo
Distinct variable groups:   ,   ,,,,   ,,,,   ,S,,,

Proof of Theorem wemappo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 simpll3 1037 . . . . . . 7
3 elmapi 7460 . . . . . . . . 9
43adantl 466 . . . . . . . 8
54ffvelrnda 6031 . . . . . . 7
6 poirr 4816 . . . . . . 7
72, 5, 6syl2anc 661 . . . . . 6
87intnanrd 917 . . . . 5
98nrexdv 2913 . . . 4
10 vex 3112 . . . . 5
11 wemapso.t . . . . . 6
1211wemaplem1 7992 . . . . 5
1310, 10, 12mp2an 672 . . . 4
149, 13sylnibr 305 . . 3
15 simpll1 1035 . . . . 5
16 simplr1 1038 . . . . 5
17 simplr2 1039 . . . . 5
18 simplr3 1040 . . . . 5
19 simpll2 1036 . . . . 5
20 simpll3 1037 . . . . 5
21 simprl 756 . . . . 5
22 simprr 757 . . . . 5
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 7994 . . . 4
2423ex 434 . . 3
2514, 24ispod 4813 . 2
261, 25syl3an1 1261 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   cvv 3109   class class class wbr 4452  {copab 4509  Powpo 4803  Orwor 4804  -->wf 5589  `cfv 5593  (class class class)co 6296   cmap 7439 This theorem is referenced by:  wemapsolem  7996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-map 7441
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