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Theorem wemappo 9580
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 8874 . . . . . . . 8 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
32adantl 480 . . . . . . 7 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → 𝑎:𝐴𝐵)
43ffvelcdmda 7099 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
5 poirr 5606 . . . . . 6 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
61, 4, 5syl2anc 582 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
76intnanrd 488 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
87nrexdv 3146 . . 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 9577 . . . 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 1212 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵m 𝐴))
14 simplr2 1213 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵m 𝐴))
15 simplr3 1214 . . . 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 9579 . . 3 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2120ex 411 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2212, 21ispod 5603 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 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473   class class class wbr 5152  {copab 5214   Po wpo 5592   Or wor 5593  wf 6549  cfv 6553  (class class class)co 7426  m cmap 8851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853
This theorem is referenced by:  wemapsolem  9581
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