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Theorem wemapso 8497
Description: Construct lexicographic order on a function space based on a well-ordering of the indexes 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 3243 . 2 (𝐴𝑉𝐴 ∈ V)
2 wemapso.t . . 3 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
3 ssid 3657 . . 3 (𝐵𝑚 𝐴) ⊆ (𝐵𝑚 𝐴)
4 simp1 1081 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝐴 ∈ V)
5 weso 5134 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
653ad2ant2 1103 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7 simp3 1083 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8 simpl1 1084 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝐴 ∈ V)
9 difss 3770 . . . . . . 7 (𝑎𝑏) ⊆ 𝑎
10 dmss 5355 . . . . . . 7 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
119, 10ax-mp 5 . . . . . 6 dom (𝑎𝑏) ⊆ dom 𝑎
12 simprll 819 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵𝑚 𝐴))
13 elmapi 7921 . . . . . . . . 9 (𝑎 ∈ (𝐵𝑚 𝐴) → 𝑎:𝐴𝐵)
1412, 13syl 17 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎:𝐴𝐵)
15 ffn 6083 . . . . . . . 8 (𝑎:𝐴𝐵𝑎 Fn 𝐴)
1614, 15syl 17 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎 Fn 𝐴)
17 fndm 6028 . . . . . . 7 (𝑎 Fn 𝐴 → dom 𝑎 = 𝐴)
1816, 17syl 17 . . . . . 6 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom 𝑎 = 𝐴)
1911, 18syl5sseq 3686 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
208, 19ssexd 4838 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
21 simpl2 1085 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑅 We 𝐴)
22 wefr 5133 . . . . 5 (𝑅 We 𝐴𝑅 Fr 𝐴)
2321, 22syl 17 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑅 Fr 𝐴)
24 simprr 811 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
25 simprlr 820 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵𝑚 𝐴))
26 elmapi 7921 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 𝐴) → 𝑏:𝐴𝐵)
2725, 26syl 17 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏:𝐴𝐵)
28 ffn 6083 . . . . . . . 8 (𝑏:𝐴𝐵𝑏 Fn 𝐴)
2927, 28syl 17 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
30 fndmdifeq0 6363 . . . . . . 7 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
3116, 29, 30syl2anc 694 . . . . . 6 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
3231necon3bid 2867 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) ≠ ∅ ↔ 𝑎𝑏))
3324, 32mpbird 247 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ≠ ∅)
34 fri 5105 . . . 4 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
3520, 23, 19, 33, 34syl22anc 1367 . . 3 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
362, 3, 4, 6, 7, 35wemapsolem 8496 . 2 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))
371, 36syl3an1 1399 1 ((𝐴𝑉𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948   class class class wbr 4685  {copab 4745   Or wor 5063   Fr wfr 5099   We wwe 5101  dom cdm 5143   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by:  opsrtoslem2  19533  wepwso  37930
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