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Theorem wemapso 8400
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 3198 . 2 (𝐴𝑉𝐴 ∈ V)
2 wemapso.t . . 3 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
3 ssid 3603 . . 3 (𝐵𝑚 𝐴) ⊆ (𝐵𝑚 𝐴)
4 simp1 1059 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝐴 ∈ V)
5 weso 5065 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
653ad2ant2 1081 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7 simp3 1061 . . 3 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8 simpl1 1062 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝐴 ∈ V)
9 difss 3715 . . . . . . 7 (𝑎𝑏) ⊆ 𝑎
10 dmss 5283 . . . . . . 7 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
119, 10ax-mp 5 . . . . . 6 dom (𝑎𝑏) ⊆ dom 𝑎
12 simprll 801 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵𝑚 𝐴))
13 elmapi 7823 . . . . . . . . 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 6002 . . . . . . . 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 5948 . . . . . . 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 3632 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
208, 19ssexd 4765 . . . 4 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
21 simpl2 1063 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑅 We 𝐴)
22 wefr 5064 . . . . 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 795 . . . . 5 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
25 simprlr 802 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵𝑚 𝐴))
26 elmapi 7823 . . . . . . . . 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 6002 . . . . . . . 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 6279 . . . . . . 7 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
3116, 29, 30syl2anc 692 . . . . . 6 (((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
3231necon3bid 2834 . . . . 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 5036 . . . 4 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
3520, 23, 19, 33, 34syl22anc 1324 . . 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 8399 . 2 ((𝐴 ∈ V ∧ 𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))
371, 36syl3an1 1356 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  c0 3891   class class class wbr 4613  {copab 4672   Or wor 4994   Fr wfr 5030   We wwe 5032  dom cdm 5074   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by:  opsrtoslem2  19404  wepwso  37093
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