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Theorem ltbval 19234
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 3180 . . . 4 (𝑇𝑊𝑇 ∈ V)
5 elex 3180 . . . 4 (𝐼𝑉𝐼 ∈ V)
6 simpr 475 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑖 = 𝐼)
76oveq2d 6539 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → (ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼))
8 rabeq 3161 . . . . . . . . . 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 2657 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
1211sseq2d 3591 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13 simpl 471 . . . . . . . . . . . 12 ((𝑟 = 𝑇𝑖 = 𝐼) → 𝑟 = 𝑇)
1413breqd 4584 . . . . . . . . . . 11 ((𝑟 = 𝑇𝑖 = 𝐼) → (𝑧𝑟𝑤𝑧𝑇𝑤))
1514imbi1d 329 . . . . . . . . . 10 ((𝑟 = 𝑇𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
166, 15raleqbidv 3124 . . . . . . . . 9 ((𝑟 = 𝑇𝑖 = 𝐼) → (∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))
1716anbi2d 735 . . . . . . . 8 ((𝑟 = 𝑇𝑖 = 𝐼) → (((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
186, 17rexeqbidv 3125 . . . . . . 7 ((𝑟 = 𝑇𝑖 = 𝐼) → (∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
1912, 18anbi12d 742 . . . . . 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 4638 . . . . 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 19122 . . . . 5 <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
22 vex 3171 . . . . . . . . 9 𝑥 ∈ V
23 vex 3171 . . . . . . . . 9 𝑦 ∈ V
2422, 23prss 4286 . . . . . . . 8 ((𝑥𝐷𝑦𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
2524anbi1i 726 . . . . . . 7 (((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
2625opabbii 4639 . . . . . 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 6551 . . . . . . . . 9 (ℕ0𝑚 𝐼) ∈ V
2810, 27rabex2 4733 . . . . . . . 8 𝐷 ∈ V
2928, 28xpex 6833 . . . . . . 7 (𝐷 × 𝐷) ∈ V
30 opabssxp 5102 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ⊆ (𝐷 × 𝐷)
3129, 30ssexi 4722 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐷𝑦𝐷) ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3226, 31eqeltrri 2680 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))} ∈ V
3320, 21, 32ovmpt2a 6663 . . . 4 ((𝑇 ∈ V ∧ 𝐼 ∈ V) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
344, 5, 33syl2an 492 . . 3 ((𝑇𝑊𝐼𝑉) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
352, 3, 34syl2anc 690 . 2 (𝜑 → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
361, 35syl5eq 2651 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 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  wss 3535  {cpr 4122   class class class wbr 4573  {copab 4632   × cxp 5022  ccnv 5023  cima 5027  cfv 5786  (class class class)co 6523  𝑚 cmap 7717  Fincfn 7814   < clt 9926  cn 10863  0cn0 11135   <bag cltb 19117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-ltbag 19122
This theorem is referenced by:  ltbwe  19235
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