Theorem fineqvnttrclselem2 35227

Description: Lemma for fineqvnttrclse 35229. (Contributed by BTernaryTau, 12-Jan-2026.)

Hypothesis

Ref	Expression
fineqvnttrclselem2.1	⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})

Assertion

Ref	Expression
fineqvnttrclselem2	⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)

Distinct variable groups:   𝑣,𝐵   𝐹,𝑑   𝑣,𝑁   𝐴,𝑑,𝑣   𝐵,𝑑

Allowed substitution hints:   𝐹(𝑣)   𝑁(𝑑)

Proof of Theorem fineqvnttrclselem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4081	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		elnn 7817	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑁 ∈ ω)
3	2	ancoms 458	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
4	1, 3	sylan 580	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
5	4	3adant3 1132	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝑁 ∈ ω)
6		fineqvnttrclselem2.1	. . . . . 6 ⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})
7		oveq1 7363	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝑣 +o 𝑑) = (𝐴 +o 𝑑))
8	7	eqeq1d 2736	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → ((𝑣 +o 𝑑) = 𝐵 ↔ (𝐴 +o 𝑑) = 𝐵))
9	8	rabbidv 3404	. . . . . . 7 ⊢ (𝑣 = 𝐴 → {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
10	9	unieqd 4874	. . . . . 6 ⊢ (𝑣 = 𝐴 → ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
11		simp3 1138	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ suc suc 𝑁)
12		fineqvnttrclselem1 35226	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
14	6, 10, 11, 13	fvmptd3 6962	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
15	5, 14	syld3an2 1413	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
16		nnon 7812	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
17	1, 16	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ On)
18		onelon 6340	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ On)
19		onsuc 7753	. . . . . . . . 9 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
21	17, 20	sylan 580	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
22		onsuc 7753	. . . . . . . 8 ⊢ (suc 𝑁 ∈ On → suc suc 𝑁 ∈ On)
23		onelon 6340	. . . . . . . 8 ⊢ ((suc suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
24	22, 23	sylan 580	. . . . . . 7 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
25	21, 24	stoic3 1777	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
26	17	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐵 ∈ On)
27		simp3 1138	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ suc suc 𝑁)
28		simpl 482	. . . . . . . . . . 11 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ∈ On)
29	24, 28	jca 511	. . . . . . . . . 10 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
30	21, 29	stoic3 1777	. . . . . . . . 9 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
31		onsssuc 6407	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑁 ∈ On) → (𝐴 ⊆ suc 𝑁 ↔ 𝐴 ∈ suc suc 𝑁))
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ⊆ suc 𝑁 ↔ 𝐴 ∈ suc suc 𝑁))
33	27, 32	mpbird 257	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ suc 𝑁)
34		nnord 7814	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → Ord 𝐵)
35		ordsucss 7758	. . . . . . . . . 10 ⊢ (Ord 𝐵 → (𝑁 ∈ 𝐵 → suc 𝑁 ⊆ 𝐵))
36	1, 34, 35	3syl 18	. . . . . . . . 9 ⊢ (𝐵 ∈ (ω ∖ 1o) → (𝑁 ∈ 𝐵 → suc 𝑁 ⊆ 𝐵))
37	36	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ⊆ 𝐵)
38	37	3adant3 1132	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ⊆ 𝐵)
39	33, 38	sstrd 3942	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ 𝐵)
40		oawordeu 8480	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
41	25, 26, 39, 40	syl21anc 837	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
42		reusn 4682	. . . . . 6 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥})
43		unieq 4872	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ {𝑥})
44		unisnv 4881	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
45	43, 44	eqtrdi 2785	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = 𝑥)
46		vsnid 4618	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
47		eleq2 2823	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → (𝑥 ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ↔ 𝑥 ∈ {𝑥}))
48	46, 47	mpbiri 258	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → 𝑥 ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
49	45, 48	eqeltrd 2834	. . . . . . 7 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
50	49	exlimiv 1931	. . . . . 6 ⊢ (∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
51	42, 50	sylbi 217	. . . . 5 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
52	41, 51	syl 17	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
53	15, 52	eqeltrd 2834	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
54		oveq2 7364	. . . . 5 ⊢ (𝑑 = (𝐹‘𝐴) → (𝐴 +o 𝑑) = (𝐴 +o (𝐹‘𝐴)))
55	54	eqeq1d 2736	. . . 4 ⊢ (𝑑 = (𝐹‘𝐴) → ((𝐴 +o 𝑑) = 𝐵 ↔ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
56	55	elrab 3644	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ↔ ((𝐹‘𝐴) ∈ On ∧ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
57	53, 56	sylib 218	. 2 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ((𝐹‘𝐴) ∈ On ∧ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
58	57	simprd 495	1 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!wreu 3346  {crab 3397   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  ∪ cuni 4861   ↦ cmpt 5177  Ord word 6314  Oncon0 6315  suc csuc 6317  ‘cfv 6490  (class class class)co 7356  ωcom 7806  1_oc1o 8388   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-en 8882  df-fin 8885

This theorem is referenced by:  fineqvnttrclselem3  35228