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Theorem ltbval 22079
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 3499 . . . 4 (𝑇𝑊𝑇 ∈ V)
5 elex 3499 . . . 4 (𝐼𝑉𝐼 ∈ V)
6 simpr 484 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑖 = 𝐼)
76oveq2d 7447 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
8 rabeq 3448 . . . . . . . . . 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, 10eqtr4di 2793 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
1211sseq2d 4028 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13 simpl 482 . . . . . . . . . . . 12 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑟 = 𝑇)
1413breqd 5159 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → (𝑧𝑟𝑤𝑧𝑇𝑤))
1514imbi1d 341 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
166, 15raleqbidv 3344 . . . . . . . . 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 3345 . . . . . . 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 5214 . . . . 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 21950 . . . . 5 <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
22 vex 3482 . . . . . . . . 9 𝑥 ∈ V
23 vex 3482 . . . . . . . . 9 𝑦 ∈ V
2422, 23prss 4825 . . . . . . . 8 ((𝑥𝐷𝑦𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
2524anbi1i 624 . . . . . . 7 (((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
2625opabbii 5215 . . . . . 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 7464 . . . . . . . . 9 (ℕ0m 𝐼) ∈ V
2810, 27rabex2 5347 . . . . . . . 8 𝐷 ∈ V
2928, 28xpex 7772 . . . . . . 7 (𝐷 × 𝐷) ∈ V
30 opabssxp 5781 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ⊆ (𝐷 × 𝐷)
3129, 30ssexi 5328 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3226, 31eqeltrri 2836 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3320, 21, 32ovmpoa 7588 . . . 4 ((𝑇 ∈ V ∧ 𝐼 ∈ V) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
344, 5, 33syl2an 596 . . 3 ((𝑇𝑊𝐼𝑉) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
352, 3, 34syl2anc 584 . 2 (𝜑 → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
361, 35eqtrid 2787 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 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  {cpr 4633   class class class wbr 5148  {copab 5210   × cxp 5687  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984   < clt 11293  cn 12264  0cn0 12524   <bag cltb 21945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ltbag 21950
This theorem is referenced by:  ltbwe  22080
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