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Theorem wemappo 8401
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 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 3198 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll3 1100 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
3 elmapi 7826 . . . . . . . . 9 (𝑎 ∈ (𝐵𝑚 𝐴) → 𝑎:𝐴𝐵)
43adantl 482 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → 𝑎:𝐴𝐵)
54ffvelrnda 6317 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
6 poirr 5008 . . . . . . 7 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
72, 5, 6syl2anc 692 . . . . . 6 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
87intnanrd 962 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
98nrexdv 2995 . . . 4 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
10 vex 3189 . . . . 5 𝑎 ∈ V
11 wemapso.t . . . . . 6 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
1211wemaplem1 8398 . . . . 5 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1310, 10, 12mp2an 707 . . . 4 (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
149, 13sylnibr 319 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ 𝑎𝑇𝑎)
15 simpll1 1098 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝐴 ∈ V)
16 simplr1 1101 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵𝑚 𝐴))
17 simplr2 1102 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵𝑚 𝐴))
18 simplr3 1103 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵𝑚 𝐴))
19 simpll2 1099 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
20 simpll3 1100 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
21 simprl 793 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
22 simprr 795 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 8400 . . . 4 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2423ex 450 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2514, 24ispod 5005 . 2 ((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
261, 25syl3an1 1356 1 ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186   class class class wbr 4615  {copab 4674   Po wpo 4995   Or wor 4996  wf 5845  cfv 5849  (class class class)co 6607  𝑚 cmap 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-po 4997  df-so 4998  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807
This theorem is referenced by:  wemapsolem  8402
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