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Theorem wemappo 9526
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.) (Revised by AV, 21-Jul-2024.)

Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemappo ((𝑅 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)

Proof of Theorem wemappo
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 774 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
2 elmapi 8826 . . . . . . . 8 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
32adantl 482 . . . . . . 7 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → 𝑎:𝐴𝐵)
43ffvelcdmda 7071 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
5 poirr 5593 . . . . . 6 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
61, 4, 5syl2anc 584 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
76intnanrd 490 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
87nrexdv 3148 . . 3 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → ¬ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
9 wemapso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
109wemaplem1 9523 . . . 4 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1110el2v 3481 . . 3 (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
128, 11sylnibr 328 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → ¬ 𝑎𝑇𝑎)
13 simplr1 1215 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵m 𝐴))
14 simplr2 1216 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵m 𝐴))
15 simplr3 1217 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵m 𝐴))
16 simplll 773 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
17 simpllr 774 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
18 simprl 769 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
19 simprr 771 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
209, 13, 14, 15, 16, 17, 18, 19wemaplem3 9525 . . 3 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2120ex 413 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2212, 21ispod 5590 1 ((𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473   class class class wbr 5141  {copab 5203   Po wpo 5579   Or wor 5580  wf 6528  cfv 6532  (class class class)co 7393  m cmap 8803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-map 8805
This theorem is referenced by:  wemapsolem  9527
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