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

Theorem wemapso 8806
 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.)
Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemapso ((𝐴𝑉𝑅 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)   𝑉(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 elex 3430 . 2 (𝐴𝑉𝐴 ∈ V)
2 wemapso.t . . 3 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
3 ssid 3878 . . 3 (𝐵𝑚 𝐴) ⊆ (𝐵𝑚 𝐴)
4 simp1 1116 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝐴 ∈ V)
5 weso 5395 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
653ad2ant2 1114 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7 simp3 1118 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8 simpl1 1171 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝐴 ∈ V)
9 difss 3997 . . . . . . 7 (𝑎𝑏) ⊆ 𝑎
10 dmss 5618 . . . . . . 7 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
119, 10ax-mp 5 . . . . . 6 dom (𝑎𝑏) ⊆ dom 𝑎
12 simprll 766 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵𝑚 𝐴))
13 elmapi 8224 . . . . . . 7 (𝑎 ∈ (𝐵𝑚 𝐴) → 𝑎:𝐴𝐵)
1412, 13syl 17 . . . . . 6 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎:𝐴𝐵)
1511, 14fssdm 6358 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
168, 15ssexd 5082 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
17 simpl2 1172 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑅 We 𝐴)
18 wefr 5394 . . . . 5 (𝑅 We 𝐴𝑅 Fr 𝐴)
1917, 18syl 17 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑅 Fr 𝐴)
20 simprr 760 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
2114ffnd 6343 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎 Fn 𝐴)
22 simprlr 767 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵𝑚 𝐴))
23 elmapi 8224 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 𝐴) → 𝑏:𝐴𝐵)
2422, 23syl 17 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏:𝐴𝐵)
2524ffnd 6343 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
26 fndmdifeq0 6637 . . . . . . 7 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2721, 25, 26syl2anc 576 . . . . . 6 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2827necon3bid 3008 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) ≠ ∅ ↔ 𝑎𝑏))
2920, 28mpbird 249 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ≠ ∅)
30 fri 5366 . . . 4 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
3116, 19, 15, 29, 30syl22anc 826 . . 3 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
322, 3, 4, 6, 7, 31wemapsolem 8805 . 2 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))
331, 32syl3an1 1143 1 ((𝐴𝑉𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048   ≠ wne 2964  ∀wral 3085  ∃wrex 3086  Vcvv 3412   ∖ cdif 3825   ⊆ wss 3828  ∅c0 4177   class class class wbr 4927  {copab 4989   Or wor 5322   Fr wfr 5360   We wwe 5362  dom cdm 5404   Fn wfn 6181  ⟶wf 6182  ‘cfv 6186  (class class class)co 6974   ↑𝑚 cmap 8202 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fv 6194  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7498  df-2nd 7499  df-map 8204 This theorem is referenced by:  opsrtoslem2  19972  wepwso  39017
 Copyright terms: Public domain W3C validator