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

Theorem wemapso 9552
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 4004 . 2 (𝐵m 𝐴) ⊆ (𝐵m 𝐴)
3 weso 5667 . . 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 3477 . . . . . 6 𝑎 ∈ V
76difexi 5328 . . . . 5 (𝑎𝑏) ∈ V
87dmex 7906 . . . 4 dom (𝑎𝑏) ∈ V
98a1i 11 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
10 wefr 5666 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
1110ad2antrr 723 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑅 Fr 𝐴)
12 difss 4131 . . . . 5 (𝑎𝑏) ⊆ 𝑎
13 dmss 5902 . . . . 5 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
1412, 13ax-mp 5 . . . 4 dom (𝑎𝑏) ⊆ dom 𝑎
15 simprll 776 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵m 𝐴))
16 elmapi 8849 . . . . 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 6737 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
19 simprr 770 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
2017ffnd 6718 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 Fn 𝐴)
21 simprlr 777 . . . . . . . 8 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵m 𝐴))
22 elmapi 8849 . . . . . . . 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 6718 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
25 fndmdifeq0 7045 . . . . . 6 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2620, 24, 25syl2anc 583 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2726necon3bid 2984 . . . 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 5636 . . 3 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
309, 11, 18, 28, 29syl22anc 836 . 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 9551 1 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  wrex 3069  Vcvv 3473  cdif 3945  wss 3948  c0 4322   class class class wbr 5148  {copab 5210   Or wor 5587   Fr wfr 5628   We wwe 5630  dom cdm 5676   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  m cmap 8826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828
This theorem is referenced by:  opsrtoslem2  21841  wepwso  42100
  Copyright terms: Public domain W3C validator