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Theorem wemappo 9001
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 3510 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll3 1206 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
3 elmapi 8417 . . . . . . . . 9 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
43adantl 482 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → 𝑎:𝐴𝐵)
54ffvelrnda 6843 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
6 poirr 5478 . . . . . . 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 3267 . . . 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 8998 . . . . 5 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1211el2v 3499 . . . 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 1204 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝐴 ∈ V)
15 simplr1 1207 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵m 𝐴))
16 simplr2 1208 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵m 𝐴))
17 simplr3 1209 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵m 𝐴))
18 simpll2 1205 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
19 simpll3 1206 . . . . 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 9000 . . . 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 5475 . 2 ((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵m 𝐴))
251, 24syl3an1 1155 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 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492   class class class wbr 5057  {copab 5119   Po wpo 5465   Or wor 5466  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by:  wemapsolem  9002
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