Theorem r1filimi 35246

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 3292	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
2		eleq1 2824	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
3	1, 2	imbi12d 344	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
4	3	imbi2d 340	. . . . 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 9690	. . . . . . . . . 10 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 483	. . . . . . . . 9 ⊢ Fun 𝑅1
7		eluniima 7205	. . . . . . . . 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 6384	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
10		ordsson 7737	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → 𝐵 ⊆ On)
12	11	sseld 3920	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	12	anim1d 612	. . . . . . . . 9 ⊢ (Lim 𝐵 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦))))
14	13	reximdv2 3147	. . . . . . . 8 ⊢ (Lim 𝐵 → (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
15	8, 14	biimtrid 242	. . . . . . 7 ⊢ (Lim 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
16	15	ralimdv 3151	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
17		vex 3433	. . . . . . . 8 ⊢ 𝑎 ∈ V
18	17	tz9.12 9714	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
19		eluniima 7205	. . . . . . . 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 234	. . . . . 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 3499	. . . 4 ⊢ (𝐴 ∈ Fin → (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
24	23	impcomd 411	. . 3 ⊢ (𝐴 ∈ Fin → ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)))
25	24	3impib 1117	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
26		simp3 1139	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → Lim 𝐵)
27		simp1 1137	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ Fin)
28		eluniima 7205	. . . . . . . . 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 3062	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)))
31		rankr1ai 9722	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝑧)
32		ordtr1 6367	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ 𝑧 ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
33	31, 32	sylani 605	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ((𝑥 ∈ (𝑅1‘𝑧) ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
34	33	ancomsd 465	. . . . . . . . . 10 ⊢ (Ord 𝐵 → ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
35	34	exlimdv 1935	. . . . . . . . 9 ⊢ (Ord 𝐵 → (∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
36	30, 35	biimtrid 242	. . . . . . . 8 ⊢ (Ord 𝐵 → (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝐵))
37	29, 36	biimtrid 242	. . . . . . 7 ⊢ (Ord 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → (rank‘𝑥) ∈ 𝐵))
38	37	ralimdv 3151	. . . . . 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 407	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
41	40	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
42		rankfilimbi 35244	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
43	27, 25, 41, 26, 42	syl22anc 839	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
44		fveq2 6840	. . . . 5 ⊢ (𝑤 = suc (rank‘𝐴) → (𝑅1‘𝑤) = (𝑅1‘suc (rank‘𝐴)))
45	44	eleq2d 2822	. . . 4 ⊢ (𝑤 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑤) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
46		limsuc 7800	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
47	46	biimpa 476	. . . . 5 ⊢ ((Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
48	47	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
49		rankidb 9724	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
50	49	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
51	45, 48, 50	rspcedvdw 3567	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
52		eluniima 7205	. . . 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 234	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
55	25, 26, 43, 54	syl3anc 1374	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  ∪ cuni 4850  dom cdm 5631   “ cima 5634  Ord word 6322  Oncon0 6323  Lim wlim 6324  suc csuc 6325  Fun wfun 6492  ‘cfv 6498  Fincfn 8893  𝑅₁cr1 9686  rankcrnk 9687

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-en 8894  df-dom 8895  df-fin 8897  df-r1 9688  df-rank 9689

This theorem is referenced by:  r1filim  35247  r1omhf  35249