Theorem r1filimi 35291

Description: If all elements in a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

Ref	Expression
r1filimi	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem r1filimi

Dummy variables 𝑤 𝑎 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3295	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
2		eleq1 2828	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
3	1, 2	imbi12d 345	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
4	3	imbi2d 341	. . . . 5 ⊢ (𝑎 = 𝐴 → ((Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On))) ↔ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)))))
5		r1funlim 9688	. . . . . . . . . 10 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 484	. . . . . . . . 9 ⊢ Fun 𝑅1
7		eluniima 7201	. . . . . . . . 9 ⊢ (Fun 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦)))
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦))
9		limord 6378	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
10		ordsson 7733	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → 𝐵 ⊆ On)
12	11	sseld 3921	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	12	anim1d 617	. . . . . . . . 9 ⊢ (Lim 𝐵 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦))))
14	13	reximdv2 3150	. . . . . . . 8 ⊢ (Lim 𝐵 → (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
15	8, 14	biimtrid 243	. . . . . . 7 ⊢ (Lim 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
16	15	ralimdv 3154	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
17		vex 3436	. . . . . . . 8 ⊢ 𝑎 ∈ V
18	17	tz9.12 9712	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
19		eluniima 7201	. . . . . . . 8 ⊢ (Fun 𝑅1 → (𝑎 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦)))
20	6, 19	ax-mp 5	. . . . . . 7 ⊢ (𝑎 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
21	18, 20	sylibr 235	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑎 ∈ ∪ (𝑅1 “ On))
22	16, 21	syl6 35	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On)))
23	4, 22	vtoclg 3502	. . . 4 ⊢ (𝐴 ∈ Fin → (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
24	23	impcomd 412	. . 3 ⊢ (𝐴 ∈ Fin → ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)))
25	24	3impib 1122	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
26		simp3 1144	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → Lim 𝐵)
27		simp1 1142	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ Fin)
28		eluniima 7201	. . . . . . . . 9 ⊢ (Fun 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧)))
29	6, 28	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧))
30		df-rex 3065	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)))
31		rankr1ai 9720	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝑧)
32		ordtr1 6361	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ 𝑧 ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
33	31, 32	sylani 610	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ((𝑥 ∈ (𝑅1‘𝑧) ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
34	33	ancomsd 466	. . . . . . . . . 10 ⊢ (Ord 𝐵 → ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
35	34	exlimdv 1940	. . . . . . . . 9 ⊢ (Ord 𝐵 → (∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
36	30, 35	biimtrid 243	. . . . . . . 8 ⊢ (Ord 𝐵 → (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝐵))
37	29, 36	biimtrid 243	. . . . . . 7 ⊢ (Ord 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → (rank‘𝑥) ∈ 𝐵))
38	37	ralimdv 3154	. . . . . 6 ⊢ (Ord 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
39	9, 38	syl 17	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
40	39	impcom 408	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
41	40	3adant1 1136	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
42		rankfilimbi 35289	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
43	27, 25, 41, 26, 42	syl22anc 844	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
44		fveq2 6834	. . . . 5 ⊢ (𝑤 = suc (rank‘𝐴) → (𝑅1‘𝑤) = (𝑅1‘suc (rank‘𝐴)))
45	44	eleq2d 2826	. . . 4 ⊢ (𝑤 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑤) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
46		limsuc 7796	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
47	46	biimpa 477	. . . . 5 ⊢ ((Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
48	47	3adant1 1136	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
49		rankidb 9722	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
50	49	3ad2ant1 1139	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
51	45, 48, 50	rspcedvdw 3570	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
52		eluniima 7201	. . . 4 ⊢ (Fun 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤)))
53	6, 52	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
54	51, 53	sylibr 235	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
55	25, 26, 43, 54	syl3anc 1379	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ⊆ wss 3890  ∪ cuni 4845  dom cdm 5625   “ cima 5628  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  Fun wfun 6486  ‘cfv 6492  Fincfn 8890  𝑅₁cr1 9684  rankcrnk 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-en 8891  df-dom 8892  df-fin 8894  df-r1 9686  df-rank 9687

This theorem is referenced by:  r1filim  35292  r1omhf  35294