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Theorem ltbval 20228
Description: Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
ltbval.c 𝐶 = (𝑇 <bag 𝐼)
ltbval.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
ltbval.i (𝜑𝐼𝑉)
ltbval.t (𝜑𝑇𝑊)
Assertion
Ref Expression
ltbval (𝜑𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
Distinct variable groups:   𝑥,𝑦,𝐷   𝑤,,𝑥,𝑦,𝑧,𝐼   𝜑,,𝑥,𝑦   𝑤,𝑇,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐶(𝑥,𝑦,𝑧,𝑤,)   𝐷(𝑧,𝑤,)   𝑇()   𝑉(𝑥,𝑦,𝑧,𝑤,)   𝑊(𝑥,𝑦,𝑧,𝑤,)

Proof of Theorem ltbval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltbval.c . 2 𝐶 = (𝑇 <bag 𝐼)
2 ltbval.t . . 3 (𝜑𝑇𝑊)
3 ltbval.i . . 3 (𝜑𝐼𝑉)
4 elex 3491 . . . 4 (𝑇𝑊𝑇 ∈ V)
5 elex 3491 . . . 4 (𝐼𝑉𝐼 ∈ V)
6 simpr 487 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑖 = 𝐼)
76oveq2d 7149 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
8 rabeq 3462 . . . . . . . . . 10 ((ℕ0m 𝑖) = (ℕ0m 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
97, 8syl 17 . . . . . . . . 9 ((𝑟 = 𝑇𝑖 = 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10 ltbval.d . . . . . . . . 9 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
119, 10syl6eqr 2873 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
1211sseq2d 3978 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13 simpl 485 . . . . . . . . . . . 12 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑟 = 𝑇)
1413breqd 5053 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → (𝑧𝑟𝑤𝑧𝑇𝑤))
1514imbi1d 344 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
166, 15raleqbidv 3388 . . . . . . . . 9 ((𝑟 = 𝑇𝑖 = 𝐼) → (∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
1716anbi2d 630 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → (((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
186, 17rexeqbidv 3389 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → (∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
1912, 18anbi12d 632 . . . . . 6 ((𝑟 = 𝑇𝑖 = 𝐼) → (({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))))
2019opabbidv 5108 . . . . 5 ((𝑟 = 𝑇𝑖 = 𝐼) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
21 df-ltbag 20115 . . . . 5 <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
22 vex 3476 . . . . . . . . 9 𝑥 ∈ V
23 vex 3476 . . . . . . . . 9 𝑦 ∈ V
2422, 23prss 4729 . . . . . . . 8 ((𝑥𝐷𝑦𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
2524anbi1i 625 . . . . . . 7 (((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
2625opabbii 5109 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))}
27 ovex 7166 . . . . . . . . 9 (ℕ0m 𝐼) ∈ V
2810, 27rabex2 5213 . . . . . . . 8 𝐷 ∈ V
2928, 28xpex 7454 . . . . . . 7 (𝐷 × 𝐷) ∈ V
30 opabssxp 5619 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ⊆ (𝐷 × 𝐷)
3129, 30ssexi 5202 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3226, 31eqeltrri 2908 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3320, 21, 32ovmpoa 7282 . . . 4 ((𝑇 ∈ V ∧ 𝐼 ∈ V) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
344, 5, 33syl2an 597 . . 3 ((𝑇𝑊𝐼𝑉) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
352, 3, 34syl2anc 586 . 2 (𝜑 → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
361, 35syl5eq 2867 1 (𝜑𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125  wrex 3126  {crab 3129  Vcvv 3473  wss 3913  {cpr 4545   class class class wbr 5042  {copab 5104   × cxp 5529  ccnv 5530  cima 5534  cfv 6331  (class class class)co 7133  m cmap 8384  Fincfn 8487   < clt 10653  cn 11616  0cn0 11876   <bag cltb 20110
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-ltbag 20115
This theorem is referenced by:  ltbwe  20229
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