Theorem fineqvnttrclselem2 35278

Description: Lemma for fineqvnttrclse 35280. (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 4083	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		elnn 7819	. . . . . . . 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 7365	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝑣 +o 𝑑) = (𝐴 +o 𝑑))
8	7	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → ((𝑣 +o 𝑑) = 𝐵 ↔ (𝐴 +o 𝑑) = 𝐵))
9	8	rabbidv 3406	. . . . . . 7 ⊢ (𝑣 = 𝐴 → {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵} = {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
10	9	unieqd 4876	. . . . . 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 35277	. . . . . . 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 6964	. . . . 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 7814	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
17	1, 16	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ On)
18		onelon 6342	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ On)
19		onsuc 7755	. . . . . . . . 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 7755	. . . . . . . 8 ⊢ (suc 𝑁 ∈ On → suc suc 𝑁 ∈ On)
23		onelon 6342	. . . . . . . 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 6409	. . . . . . . . 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 7816	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → Ord 𝐵)
35		ordsucss 7760	. . . . . . . . . 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 3944	. . . . . 6 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → 𝐴 ⊆ 𝐵)
40		oawordeu 8482	. . . . . 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 4684	. . . . . 6 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑥{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥})
43		unieq 4874	. . . . . . . . 9 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ {𝑥})
44		unisnv 4883	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
45	43, 44	eqtrdi 2787	. . . . . . . 8 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑥} → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = 𝑥)
46		vsnid 4620	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
47		eleq2 2825	. . . . . . . . 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 2836	. . . . . . 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 2836	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐹‘𝐴) ∈ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵})
54		oveq2 7366	. . . . 5 ⊢ (𝑑 = (𝐹‘𝐴) → (𝐴 +o 𝑑) = (𝐴 +o (𝐹‘𝐴)))
55	54	eqeq1d 2738	. . . 4 ⊢ (𝑑 = (𝐹‘𝐴) → ((𝐴 +o 𝑑) = 𝐵 ↔ (𝐴 +o (𝐹‘𝐴)) = 𝐵))
56	55	elrab 3646	. . 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 3348  {crab 3399   ∖ cdif 3898   ⊆ wss 3901  {csn 4580  ∪ cuni 4863   ↦ cmpt 5179  Ord word 6316  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808  1_oc1o 8390   +_o coa 8394

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-en 8884  df-fin 8887

This theorem is referenced by:  fineqvnttrclselem3  35279