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Theorem wemappo 8067
 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:   ,   ,,,,   ,,,,   ,,,,
Allowed substitution hints:   (,,)   (,,,)   (,,,)

Proof of Theorem wemappo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3090 . 2
2 simpll3 1046 . . . . . . 7
3 elmapi 7498 . . . . . . . . 9
43adantl 467 . . . . . . . 8
54ffvelrnda 6034 . . . . . . 7
6 poirr 4782 . . . . . . 7
72, 5, 6syl2anc 665 . . . . . 6
87intnanrd 925 . . . . 5
98nrexdv 2881 . . . 4
10 vex 3084 . . . . 5
11 wemapso.t . . . . . 6
1211wemaplem1 8064 . . . . 5
1310, 10, 12mp2an 676 . . . 4
149, 13sylnibr 306 . . 3
15 simpll1 1044 . . . . 5
16 simplr1 1047 . . . . 5
17 simplr2 1048 . . . . 5
18 simplr3 1049 . . . . 5
19 simpll2 1045 . . . . 5
20 simpll3 1046 . . . . 5
21 simprl 762 . . . . 5
22 simprr 764 . . . . 5
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 8066 . . . 4
2423ex 435 . . 3
2514, 24ispod 4779 . 2
261, 25syl3an1 1297 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   w3a 982   wceq 1437   wcel 1868  wral 2775  wrex 2776  cvv 3081   class class class wbr 4420  copab 4478   wpo 4769   wor 4770  wf 5594  cfv 5598  (class class class)co 6302   cmap 7477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-1st 6804  df-2nd 6805  df-map 7479 This theorem is referenced by:  wemapsolem  8068
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