MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wemapso Structured version   Visualization version   GIF version

Theorem wemapso 9589
Description: Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-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
wemapso ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (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 wemapso
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemapso.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2 ssid 4018 . 2 (𝐵m 𝐴) ⊆ (𝐵m 𝐴)
3 weso 5680 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
43adantr 480 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5 simpr 484 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6 vex 3482 . . . . . 6 𝑎 ∈ V
76difexi 5336 . . . . 5 (𝑎𝑏) ∈ V
87dmex 7932 . . . 4 dom (𝑎𝑏) ∈ V
98a1i 11 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
10 wefr 5679 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
1110ad2antrr 726 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑅 Fr 𝐴)
12 difss 4146 . . . . 5 (𝑎𝑏) ⊆ 𝑎
13 dmss 5916 . . . . 5 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
1412, 13ax-mp 5 . . . 4 dom (𝑎𝑏) ⊆ dom 𝑎
15 simprll 779 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵m 𝐴))
16 elmapi 8888 . . . . 5 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
1715, 16syl 17 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎:𝐴𝐵)
1814, 17fssdm 6756 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
19 simprr 773 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
2017ffnd 6738 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 Fn 𝐴)
21 simprlr 780 . . . . . . . 8 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵m 𝐴))
22 elmapi 8888 . . . . . . . 8 (𝑏 ∈ (𝐵m 𝐴) → 𝑏:𝐴𝐵)
2321, 22syl 17 . . . . . . 7 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏:𝐴𝐵)
2423ffnd 6738 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
25 fndmdifeq0 7064 . . . . . 6 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2620, 24, 25syl2anc 584 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2726necon3bid 2983 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) ≠ ∅ ↔ 𝑎𝑏))
2819, 27mpbird 257 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ≠ ∅)
29 fri 5646 . . 3 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
309, 11, 18, 28, 29syl22anc 839 . 2 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
311, 2, 4, 5, 30wemapsolem 9588 1 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  c0 4339   class class class wbr 5148  {copab 5210   Or wor 5596   Fr wfr 5638   We wwe 5640  dom cdm 5689   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  opsrtoslem2  22098  wepwso  43032
  Copyright terms: Public domain W3C validator