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Theorem wemappo 8849
 Description: Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. Without totality on the values or least differing indices, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemappo ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemappo
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3450 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll3 1205 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
3 elmapi 8269 . . . . . . . . 9 (𝑎 ∈ (𝐵𝑚 𝐴) → 𝑎:𝐴𝐵)
43adantl 482 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → 𝑎:𝐴𝐵)
54ffvelrnda 6707 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
6 poirr 5365 . . . . . . 7 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
72, 5, 6syl2anc 584 . . . . . 6 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
87intnanrd 490 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
98nrexdv 3230 . . . 4 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
10 wemapso.t . . . . . 6 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
1110wemaplem1 8846 . . . . 5 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1211el2v 3439 . . . 4 (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
139, 12sylnibr 330 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ 𝑎𝑇𝑎)
14 simpll1 1203 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝐴 ∈ V)
15 simplr1 1206 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵𝑚 𝐴))
16 simplr2 1207 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵𝑚 𝐴))
17 simplr3 1208 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵𝑚 𝐴))
18 simpll2 1204 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
19 simpll3 1205 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
20 simprl 767 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
21 simprr 769 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
2210, 14, 15, 16, 17, 18, 19, 20, 21wemaplem3 8848 . . . 4 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2322ex 413 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2413, 23ispod 5362 . 2 ((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
251, 24syl3an1 1154 1 ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104  Vcvv 3432   class class class wbr 4956  {copab 5018   Po wpo 5352   Or wor 5353  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ↑𝑚 cmap 8247 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-po 5354  df-so 5355  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-1st 7536  df-2nd 7537  df-map 8249 This theorem is referenced by:  wemapsolem  8850
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