Theorem fineqvnttrclselem2 35651

Description: Lemma for fineqvnttrclse 35653. (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 4078	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		elnn 7874	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑁 ∈ ω)
3	2	ancoms 464	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
4	1, 3	sylan 592	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
5	4	3adant3 1150	. . . . 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 7421	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝑣 +o 𝑑) = (𝐴 +o 𝑑))
8	7	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → ((𝑣 +o 𝑑) = 𝐵 ↔ (𝐴 +o 𝑑) = 𝐵))
9	8	rabbidv 3419	. . . . . . 7 ⊢ (𝑣 = 𝐴 → {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
10	9	unieqd 4880	. . . . . 6 ⊢ (𝑣 = 𝐴 → ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
11		simp3 1156	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ suc suc 𝑁)
12		fineqvnttrclselem1 35650	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
13	12	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
14	6, 10, 11, 13	fvmptd3 7011	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
15	5, 14	syld3an2 1438	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
16		nnon 7869	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
17	1, 16	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ On)
18		onelon 6382	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ On)
19		onsuc 7810	. . . . . . . . 9 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
20	18, 19	syl 18	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
21	17, 20	sylan 592	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
22		onsuc 7810	. . . . . . . 8 ⊢ (suc 𝑁 ∈ On → suc suc 𝑁 ∈ On)
23		onelon 6382	. . . . . . . 8 ⊢ ((suc suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
24	22, 23	sylan 592	. . . . . . 7 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
25	21, 24	stoic3 1809	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
26	17	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐵 ∈ On)
27		simp3 1156	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ suc suc 𝑁)
28		simpl 488	. . . . . . . . . . 11 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ∈ On)
29	24, 28	jca 521	. . . . . . . . . 10 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
30	21, 29	stoic3 1809	. . . . . . . . 9 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
31		onsssuc 6450	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑁 ∈ On) → (𝐴 ⊆ suc 𝑁 ↔ 𝐴 ∈ suc suc 𝑁))
32	30, 31	syl 18	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ⊆ suc 𝑁 ↔ 𝐴 ∈ suc suc 𝑁))
33	27, 32	mpbird 260	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ suc 𝑁)
34		nnord 7871	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → Ord 𝐵)
35		ordsucss 7815	. . . . . . . . . 10 ⊢ (Ord 𝐵 → (𝑁 ∈ 𝐵 → suc 𝑁 ⊆ 𝐵))
36	1, 34, 35	3syl 19	. . . . . . . . 9 ⊢ (𝐵 ∈ (ω ∖ 1o) → (𝑁 ∈ 𝐵 → suc 𝑁 ⊆ 𝐵))
37	36	imp 412	. . . . . . . 8 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ⊆ 𝐵)
38	37	3adant3 1150	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ⊆ 𝐵)
39	33, 38	sstrd 3941	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ 𝐵)
40		oawordeu 8545	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
41	25, 26, 39, 40	syl21anc 851	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
42		reusn 4688	. . . . . 6 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥})
43		unieq 4878	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ {𝑥})
44		unisnv 4887	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
45	43, 44	eqtrdi 2811	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = 𝑥)
46		vsnid 4624	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
47		eleq2 2849	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → (𝑥 ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ↔ 𝑥 ∈ {𝑥}))
48	46, 47	mpbiri 261	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → 𝑥 ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
49	45, 48	eqeltrd 2860	. . . . . . 7 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
50	49	exlimiv 1963	. . . . . 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 220	. . . . 5 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
52	41, 51	syl 18	. . . 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 2860	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
54		oveq2 7422	. . . . 5 ⊢ (𝑑 = (𝐹‘𝐴) → (𝐴 +o 𝑑) = (𝐴 +o (𝐹‘𝐴)))
55	54	eqeq1d 2762	. . . 4 ⊢ (𝑑 = (𝐹‘𝐴) → ((𝐴 +o 𝑑) = 𝐵 ↔ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
56	55	elrab 3645	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ↔ ((𝐹‘𝐴) ∈ On ∧ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
57	53, 56	sylib 221	. 2 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ((𝐹‘𝐴) ∈ On ∧ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
58	57	simprd 501	1 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!wreu 3363  {crab 3412   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  Ord word 6356  Oncon0 6357  suc csuc 6359  ‘cfv 6533  (class class class)co 7414  ωcom 7863  1_oc1o 8451   +_o coa 8455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-oadd 8462  df-en 8956  df-fin 8959

This theorem is used by:  fineqvnttrclselem3  35652