Theorem ltbval 21988

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 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
ltbval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
ltbval.t	⊢ (𝜑 → 𝑇 ∈ 𝑊)

Assertion

Ref	Expression
ltbval	⊢ (𝜑 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})

Distinct variable groups:   𝑥,𝑦,𝐷   𝑤,ℎ,𝑥,𝑦,𝑧,𝐼   𝜑,ℎ,𝑥,𝑦   𝑤,𝑇,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐶(𝑥,𝑦,𝑧,𝑤,ℎ)   𝐷(𝑧,𝑤,ℎ)   𝑇(ℎ)   𝑉(𝑥,𝑦,𝑧,𝑤,ℎ)   𝑊(𝑥,𝑦,𝑧,𝑤,ℎ)

Proof of Theorem ltbval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltbval.c	. 2 ⊢ 𝐶 = (𝑇 <bag 𝐼)
2		ltbval.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
3		ltbval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
4		elex 3492	. . . 4 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
5		elex 3492	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
6		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
7	6	oveq2d 7442	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
8		rabeq 3445	. . . . . . . . . 10 ⊢ ((ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
10		ltbval.d	. . . . . . . . 9 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
11	9, 10	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
12	11	sseq2d 4014	. . . . . . 7 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13		simpl 481	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑇)
14	13	breqd 5163	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (𝑧𝑟𝑤 ↔ 𝑧𝑇𝑤))
15	14	imbi1d 340	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
16	6, 15	raleqbidv 3340	. . . . . . . . 9 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
17	16	anbi2d 628	. . . . . . . 8 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
18	6, 17	rexeqbidv 3341	. . . . . . 7 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
19	12, 18	anbi12d 630	. . . . . 6 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ ∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))))
20	19	opabbidv 5218	. . . . 5 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ ∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
21		df-ltbag 21852	. . . . 5 ⊢ <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ ∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
22		vex 3477	. . . . . . . . 9 ⊢ 𝑥 ∈ V
23		vex 3477	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	22, 23	prss 4828	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
25	24	anbi1i 622	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
26	25	opabbii 5219	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))}
27		ovex 7459	. . . . . . . . 9 ⊢ (ℕ0 ↑m 𝐼) ∈ V
28	10, 27	rabex2 5340	. . . . . . . 8 ⊢ 𝐷 ∈ V
29	28, 28	xpex 7761	. . . . . . 7 ⊢ (𝐷 × 𝐷) ∈ V
30		opabssxp 5774	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))} ⊆ (𝐷 × 𝐷)
31	29, 30	ssexi 5326	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))} ∈ V
32	26, 31	eqeltrri 2826	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))} ∈ V
33	20, 21, 32	ovmpoa 7582	. . . 4 ⊢ ((𝑇 ∈ V ∧ 𝐼 ∈ V) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
34	4, 5, 33	syl2an 594	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝐼 ∈ 𝑉) → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
35	2, 3, 34	syl2anc 582	. 2 ⊢ (𝜑 → (𝑇 <bag 𝐼) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
36	1, 35	eqtrid 2780	1 ⊢ (𝜑 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  {crab 3430  Vcvv 3473   ⊆ wss 3949  {cpr 4634   class class class wbr 5152  {copab 5214   × cxp 5680  ^◡ccnv 5681   “ cima 5685  ‘cfv 6553  (class class class)co 7426   ↑_m cmap 8851  Fincfn 8970   < clt 11286  ℕcn 12250  ℕ₀cn0 12510   <_bag cltb 21847

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-ltbag 21852

This theorem is referenced by:  ltbwe  21989