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Theorem ltbval 21936
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 3487 . . . 4 (𝑇𝑊𝑇 ∈ V)
5 elex 3487 . . . 4 (𝐼𝑉𝐼 ∈ V)
6 simpr 484 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑖 = 𝐼)
76oveq2d 7420 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
8 rabeq 3440 . . . . . . . . . 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 2784 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
1211sseq2d 4009 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13 simpl 482 . . . . . . . . . . . 12 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑟 = 𝑇)
1413breqd 5152 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → (𝑧𝑟𝑤𝑧𝑇𝑤))
1514imbi1d 341 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
166, 15raleqbidv 3336 . . . . . . . . 9 ((𝑟 = 𝑇𝑖 = 𝐼) → (∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
1716anbi2d 628 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → (((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
186, 17rexeqbidv 3337 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → (∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
1912, 18anbi12d 630 . . . . . 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 5207 . . . . 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 21802 . . . . 5 <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
22 vex 3472 . . . . . . . . 9 𝑥 ∈ V
23 vex 3472 . . . . . . . . 9 𝑦 ∈ V
2422, 23prss 4818 . . . . . . . 8 ((𝑥𝐷𝑦𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
2524anbi1i 623 . . . . . . 7 (((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
2625opabbii 5208 . . . . . 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 7437 . . . . . . . . 9 (ℕ0m 𝐼) ∈ V
2810, 27rabex2 5327 . . . . . . . 8 𝐷 ∈ V
2928, 28xpex 7736 . . . . . . 7 (𝐷 × 𝐷) ∈ V
30 opabssxp 5761 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ⊆ (𝐷 × 𝐷)
3129, 30ssexi 5315 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3226, 31eqeltrri 2824 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3320, 21, 32ovmpoa 7558 . . . 4 ((𝑇 ∈ V ∧ 𝐼 ∈ V) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
344, 5, 33syl2an 595 . . 3 ((𝑇𝑊𝐼𝑉) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
352, 3, 34syl2anc 583 . 2 (𝜑 → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
361, 35eqtrid 2778 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 1533  wcel 2098  wral 3055  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  {cpr 4625   class class class wbr 5141  {copab 5203   × cxp 5667  ccnv 5668  cima 5672  cfv 6536  (class class class)co 7404  m cmap 8819  Fincfn 8938   < clt 11249  cn 12213  0cn0 12473   <bag cltb 21797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-ltbag 21802
This theorem is referenced by:  ltbwe  21937
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