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Theorem wemappo 9544
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 775 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
2 elmapi 8843 . . . . . . . 8 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
32adantl 483 . . . . . . 7 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → 𝑎:𝐴𝐵)
43ffvelcdmda 7087 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
5 poirr 5601 . . . . . 6 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
61, 4, 5syl2anc 585 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
76intnanrd 491 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
87nrexdv 3150 . . 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 9541 . . . 4 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1110el2v 3483 . . 3 (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
128, 11sylnibr 329 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵m 𝐴)) → ¬ 𝑎𝑇𝑎)
13 simplr1 1216 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵m 𝐴))
14 simplr2 1217 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵m 𝐴))
15 simplr3 1218 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵m 𝐴))
16 simplll 774 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
17 simpllr 775 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
18 simprl 770 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
19 simprr 772 . . . 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 9543 . . 3 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2120ex 414 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴) ∧ 𝑐 ∈ (𝐵m 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2212, 21ispod 5598 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475   class class class wbr 5149  {copab 5211   Po wpo 5587   Or wor 5588  wf 6540  cfv 6544  (class class class)co 7409  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by:  wemapsolem  9545
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