Theorem fineqvnttrclselem2 35535

Description: Lemma for fineqvnttrclse 35537. (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 4085	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		elnn 7869	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑁 ∈ ω)
3	2	ancoms 463	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
4	1, 3	sylan 591	. . . . . 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 7417	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝑣 +o 𝑑) = (𝐴 +o 𝑑))
8	7	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → ((𝑣 +o 𝑑) = 𝐵 ↔ (𝐴 +o 𝑑) = 𝐵))
9	8	rabbidv 3423	. . . . . . 7 ⊢ (𝑣 = 𝐴 → {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
10	9	unieqd 4885	. . . . . 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 35534	. . . . . . 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 7013	. . . . 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 7864	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
17	1, 16	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ On)
18		onelon 6385	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ On)
19		onsuc 7805	. . . . . . . . 9 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
20	18, 19	syl 18	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
21	17, 20	sylan 591	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
22		onsuc 7805	. . . . . . . 8 ⊢ (suc 𝑁 ∈ On → suc suc 𝑁 ∈ On)
23		onelon 6385	. . . . . . . 8 ⊢ ((suc suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
24	22, 23	sylan 591	. . . . . . 7 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
25	21, 24	stoic3 1806	. . . . . 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 487	. . . . . . . . . . 11 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ∈ On)
29	24, 28	jca 520	. . . . . . . . . 10 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
30	21, 29	stoic3 1806	. . . . . . . . 9 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
31		onsssuc 6453	. . . . . . . . 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 7866	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → Ord 𝐵)
35		ordsucss 7810	. . . . . . . . . 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 411	. . . . . . . 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 3947	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ 𝐵)
40		oawordeu 8536	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
41	25, 26, 39, 40	syl21anc 850	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
42		reusn 4693	. . . . . 6 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥})
43		unieq 4883	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ {𝑥})
44		unisnv 4892	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
45	43, 44	eqtrdi 2814	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = 𝑥)
46		vsnid 4629	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
47		eleq2 2852	. . . . . . . . 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 2863	. . . . . . 7 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
50	49	exlimiv 1960	. . . . . 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 2863	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
54		oveq2 7418	. . . . 5 ⊢ (𝑑 = (𝐹‘𝐴) → (𝐴 +o 𝑑) = (𝐴 +o (𝐹‘𝐴)))
55	54	eqeq1d 2765	. . . 4 ⊢ (𝑑 = (𝐹‘𝐴) → ((𝐴 +o 𝑑) = 𝐵 ↔ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
56	55	elrab 3650	. . 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 500	1 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃!wreu 3367  {crab 3416   ∖ cdif 3902   ⊆ wss 3905  {csn 4589  ∪ cuni 4872   ↦ cmpt 5192  Ord word 6359  Oncon0 6360  suc csuc 6362  ‘cfv 6536  (class class class)co 7410  ωcom 7858  1_oc1o 8442   +_o coa 8446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-en 8940  df-fin 8943

This theorem is referenced by:  fineqvnttrclselem3  35536