Theorem fineqvnttrclselem2 35300

Description: Lemma for fineqvnttrclse 35302. (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 7829	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑁 ∈ ω)
3	2	ancoms 458	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
4	1, 3	sylan 581	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ω)
5	4	3adant3 1133	. . . . 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 7375	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝑣 +o 𝑑) = (𝐴 +o 𝑑))
8	7	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → ((𝑣 +o 𝑑) = 𝐵 ↔ (𝐴 +o 𝑑) = 𝐵))
9	8	rabbidv 3408	. . . . . . 7 ⊢ (𝑣 = 𝐴 → {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
10	9	unieqd 4878	. . . . . 6 ⊢ (𝑣 = 𝐴 → ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
11		simp3 1139	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ suc suc 𝑁)
12		fineqvnttrclselem1 35299	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
13	12	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
14	6, 10, 11, 13	fvmptd3 6973	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ ω ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
15	5, 14	syld3an2 1414	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) = ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
16		nnon 7824	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
17	1, 16	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ On)
18		onelon 6350	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ On)
19		onsuc 7765	. . . . . . . . 9 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
21	17, 20	sylan 581	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → suc 𝑁 ∈ On)
22		onsuc 7765	. . . . . . . 8 ⊢ (suc 𝑁 ∈ On → suc suc 𝑁 ∈ On)
23		onelon 6350	. . . . . . . 8 ⊢ ((suc suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
24	22, 23	sylan 581	. . . . . . 7 ⊢ ((suc 𝑁 ∈ On ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
25	21, 24	stoic3 1778	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ∈ On)
26	17	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐵 ∈ On)
27		simp3 1139	. . . . . . . 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 1778	. . . . . . . . 9 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 ∈ On ∧ suc 𝑁 ∈ On))
31		onsssuc 6417	. . . . . . . . 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 7826	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → Ord 𝐵)
35		ordsucss 7770	. . . . . . . . . 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 1133	. . . . . . 7 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → suc 𝑁 ⊆ 𝐵)
39	33, 38	sstrd 3946	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ 𝐵)
40		oawordeu 8492	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
41	25, 26, 39, 40	syl21anc 838	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
42		reusn 4686	. . . . . 6 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥})
43		unieq 4876	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ {𝑥})
44		unisnv 4885	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
45	43, 44	eqtrdi 2788	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = 𝑥)
46		vsnid 4622	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
47		eleq2 2826	. . . . . . . . 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 2837	. . . . . . 7 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
50	49	exlimiv 1932	. . . . . 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 2837	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
54		oveq2 7376	. . . . 5 ⊢ (𝑑 = (𝐹‘𝐴) → (𝐴 +o 𝑑) = (𝐴 +o (𝐹‘𝐴)))
55	54	eqeq1d 2739	. . . 4 ⊢ (𝑑 = (𝐹‘𝐴) → ((𝐴 +o 𝑑) = 𝐵 ↔ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
56	55	elrab 3648	. . 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!wreu 3350  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  Ord word 6324  Oncon0 6325  suc csuc 6327  ‘cfv 6500  (class class class)co 7368  ωcom 7818  1_oc1o 8400   +_o coa 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-en 8896  df-fin 8899

This theorem is referenced by:  fineqvnttrclselem3  35301