Theorem ltbval 21255

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 3449	. . . 4 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
5		elex 3449	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
6		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
7	6	oveq2d 7288	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
8		rabeq 3417	. . . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
12	11	sseq2d 3958	. . . . . . 7 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ {𝑥, 𝑦} ⊆ 𝐷))
13		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑇)
14	13	breqd 5090	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (𝑧𝑟𝑤 ↔ 𝑧𝑇𝑤))
15	14	imbi1d 342	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → ((𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
16	6, 15	raleqbidv 3335	. . . . . . . . 9 ⊢ ((𝑟 = 𝑇 ∧ 𝑖 = 𝐼) → (∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
17	16	anbi2d 629	. . . . . . . 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 3336	. . . . . . 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 631	. . . . . 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 5145	. . . . 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 21126	. . . . 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 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
23		vex 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	22, 23	prss 4759	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷)
25	24	anbi1i 624	. . . . . . 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 5146	. . . . . 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 7305	. . . . . . . . 9 ⊢ (ℕ0 ↑m 𝐼) ∈ V
28	10, 27	rabex2 5262	. . . . . . . 8 ⊢ 𝐷 ∈ V
29	28, 28	xpex 7598	. . . . . . 7 ⊢ (𝐷 × 𝐷) ∈ V
30		opabssxp 5679	. . . . . . 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 5250	. . . . . 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 2838	. . . . 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 7423	. . . 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 596	. . 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 584	. 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 2792	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 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  {crab 3070  Vcvv 3431   ⊆ wss 3892  {cpr 4569   class class class wbr 5079  {copab 5141   × cxp 5588  ^◡ccnv 5589   “ cima 5593  ‘cfv 6432  (class class class)co 7272   ↑_m cmap 8607  Fincfn 8725   < clt 11020  ℕcn 11984  ℕ₀cn0 12244   <_bag cltb 21121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-ltbag 21126

This theorem is referenced by:  ltbwe  21256