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Theorem ltbwe 19293
 Description: The finite bag order is a well-order, given a well-order of the index set. (Contributed by Mario Carneiro, 2-Jun-2015.)
Hypotheses
Ref Expression
ltbval.c 𝐶 = (𝑇 <bag 𝐼)
ltbval.d 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
ltbval.i (𝜑𝐼𝑉)
ltbval.t (𝜑𝑇𝑊)
ltbwe.w (𝜑𝑇 We 𝐼)
Assertion
Ref Expression
ltbwe (𝜑𝐶 We 𝐷)
Distinct variable groups:   ,𝐼   𝜑,
Allowed substitution hints:   𝐶()   𝐷()   𝑇()   𝑉()   𝑊()

Proof of Theorem ltbwe
Dummy variables 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
2 breq1 4586 . . . . . 6 ( = 𝑥 → ( finSupp 0 ↔ 𝑥 finSupp 0))
32cbvrabv 3172 . . . . 5 { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0} = {𝑥 ∈ (ℕ0𝑚 𝐼) ∣ 𝑥 finSupp 0}
4 ltbwe.w . . . . 5 (𝜑𝑇 We 𝐼)
5 nn0uz 11598 . . . . . 6 0 = (ℤ‘0)
6 ltweuz 12622 . . . . . . 7 < We (ℤ‘0)
7 weeq2 5027 . . . . . . 7 (ℕ0 = (ℤ‘0) → ( < We ℕ0 ↔ < We (ℤ‘0)))
86, 7mpbiri 247 . . . . . 6 (ℕ0 = (ℤ‘0) → < We ℕ0)
95, 8mp1i 13 . . . . 5 (𝜑 → < We ℕ0)
10 0nn0 11184 . . . . . 6 0 ∈ ℕ0
11 ne0i 3880 . . . . . 6 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1210, 11mp1i 13 . . . . 5 (𝜑 → ℕ0 ≠ ∅)
13 eqid 2610 . . . . 5 OrdIso(𝑇, 𝐼) = OrdIso(𝑇, 𝐼)
14 0z 11265 . . . . . . 7 0 ∈ ℤ
15 hashgval2 13028 . . . . . . 7 (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
1614, 15om2uzoi 12616 . . . . . 6 (# ↾ ω) = OrdIso( < , (ℤ‘0))
17 oieq2 8301 . . . . . . 7 (ℕ0 = (ℤ‘0) → OrdIso( < , ℕ0) = OrdIso( < , (ℤ‘0)))
185, 17ax-mp 5 . . . . . 6 OrdIso( < , ℕ0) = OrdIso( < , (ℤ‘0))
1916, 18eqtr4i 2635 . . . . 5 (# ↾ ω) = OrdIso( < , ℕ0)
20 peano1 6977 . . . . . . 7 ∅ ∈ ω
21 fvres 6117 . . . . . . 7 (∅ ∈ ω → ((# ↾ ω)‘∅) = (#‘∅))
2220, 21ax-mp 5 . . . . . 6 ((# ↾ ω)‘∅) = (#‘∅)
23 hash0 13019 . . . . . 6 (#‘∅) = 0
2422, 23eqtr2i 2633 . . . . 5 0 = ((# ↾ ω)‘∅)
251, 3, 4, 9, 12, 13, 19, 24wemapwe 8477 . . . 4 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0})
26 ltbval.d . . . . . 6 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
27 elmapfun 7767 . . . . . . . . . 10 ( ∈ (ℕ0𝑚 𝐼) → Fun )
2827adantl 481 . . . . . . . . 9 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → Fun )
29 simpr 476 . . . . . . . . 9 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → ∈ (ℕ0𝑚 𝐼))
30 c0ex 9913 . . . . . . . . . 10 0 ∈ V
3130a1i 11 . . . . . . . . 9 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → 0 ∈ V)
32 funisfsupp 8163 . . . . . . . . 9 ((Fun ∈ (ℕ0𝑚 𝐼) ∧ 0 ∈ V) → ( finSupp 0 ↔ ( supp 0) ∈ Fin))
3328, 29, 31, 32syl3anc 1318 . . . . . . . 8 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → ( finSupp 0 ↔ ( supp 0) ∈ Fin))
34 ltbval.i . . . . . . . . 9 (𝜑𝐼𝑉)
35 elmapi 7765 . . . . . . . . 9 ( ∈ (ℕ0𝑚 𝐼) → :𝐼⟶ℕ0)
36 frnnn0supp 11226 . . . . . . . . . 10 ((𝐼𝑉:𝐼⟶ℕ0) → ( supp 0) = ( “ ℕ))
3736eleq1d 2672 . . . . . . . . 9 ((𝐼𝑉:𝐼⟶ℕ0) → (( supp 0) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
3834, 35, 37syl2an 493 . . . . . . . 8 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → (( supp 0) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
3933, 38bitr2d 268 . . . . . . 7 ((𝜑 ∈ (ℕ0𝑚 𝐼)) → (( “ ℕ) ∈ Fin ↔ finSupp 0))
4039rabbidva 3163 . . . . . 6 (𝜑 → { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0})
4126, 40syl5eq 2656 . . . . 5 (𝜑𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0})
42 weeq2 5027 . . . . 5 (𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0} → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We 𝐷 ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0}))
4341, 42syl 17 . . . 4 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We 𝐷 ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We { ∈ (ℕ0𝑚 𝐼) ∣ finSupp 0}))
4425, 43mpbird 246 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We 𝐷)
45 weinxp 5109 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} We 𝐷 ↔ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)) We 𝐷)
4644, 45sylib 207 . 2 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)) We 𝐷)
47 ltbval.c . . . . 5 𝐶 = (𝑇 <bag 𝐼)
48 ltbval.t . . . . 5 (𝜑𝑇𝑊)
4947, 26, 34, 48ltbval 19292 . . . 4 (𝜑𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
50 df-xp 5044 . . . . . . 7 (𝐷 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐷𝑦𝐷)}
51 vex 3176 . . . . . . . . 9 𝑥 ∈ V
52 vex 3176 . . . . . . . . 9 𝑦 ∈ V
5351, 52prss 4291 . . . . . . . 8 ((𝑥𝐷𝑦𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
5453opabbii 4649 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐷𝑦𝐷)} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ⊆ 𝐷}
5550, 54eqtr2i 2633 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ⊆ 𝐷} = (𝐷 × 𝐷)
5655ineq1i 3772 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ⊆ 𝐷} ∩ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))}) = ((𝐷 × 𝐷) ∩ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))})
57 inopab 5174 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ⊆ 𝐷} ∩ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))}) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))}
58 incom 3767 . . . . 5 ((𝐷 × 𝐷) ∩ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))}) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷))
5956, 57, 583eqtr3i 2640 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷))
6049, 59syl6eq 2660 . . 3 (𝜑𝐶 = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)))
61 weeq1 5026 . . 3 (𝐶 = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)) → (𝐶 We 𝐷 ↔ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)) We 𝐷))
6260, 61syl 17 . 2 (𝜑 → (𝐶 We 𝐷 ↔ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ (𝐷 × 𝐷)) We 𝐷))
6346, 62mpbird 246 1 (𝜑𝐶 We 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127   class class class wbr 4583  {copab 4642   We wwe 4996   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   supp csupp 7182   ↑𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  OrdIsocoi 8297  0cc0 9815   < clt 9953  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  #chash 12979
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