Theorem r1filimi 35135

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 3290	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
2		eleq1 2821	. . . . . . 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 9666	. . . . . . . . . 10 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 483	. . . . . . . . 9 ⊢ Fun 𝑅1
7		eluniima 7190	. . . . . . . . 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 6372	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
10		ordsson 7722	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → 𝐵 ⊆ On)
12	11	sseld 3929	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	12	anim1d 611	. . . . . . . . 9 ⊢ (Lim 𝐵 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦))))
14	13	reximdv2 3143	. . . . . . . 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 3147	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
17		vex 3441	. . . . . . . 8 ⊢ 𝑎 ∈ V
18	17	tz9.12 9690	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
19		eluniima 7190	. . . . . . . 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 3508	. . . 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 1116	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
26		simp3 1138	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → Lim 𝐵)
27		simp1 1136	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ Fin)
28		eluniima 7190	. . . . . . . . 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 3058	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)))
31		rankr1ai 9698	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝑧)
32		ordtr1 6355	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ 𝑧 ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
33	31, 32	sylani 604	. . . . . . . . . . 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 1934	. . . . . . . . 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 3147	. . . . . 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 1130	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
42		rankfilimbi 35133	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
43	27, 25, 41, 26, 42	syl22anc 838	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
44		fveq2 6828	. . . . 5 ⊢ (𝑤 = suc (rank‘𝐴) → (𝑅1‘𝑤) = (𝑅1‘suc (rank‘𝐴)))
45	44	eleq2d 2819	. . . 4 ⊢ (𝑤 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑤) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
46		limsuc 7785	. . . . . 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 1130	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
49		rankidb 9700	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
50	49	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
51	45, 48, 50	rspcedvdw 3576	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
52		eluniima 7190	. . . 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 1373	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057   ⊆ wss 3898  ∪ cuni 4858  dom cdm 5619   “ cima 5622  Ord word 6310  Oncon0 6311  Lim wlim 6312  suc csuc 6313  Fun wfun 6480  ‘cfv 6486  Fincfn 8875  𝑅₁cr1 9662  rankcrnk 9663

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-en 8876  df-dom 8877  df-fin 8879  df-r1 9664  df-rank 9665

This theorem is referenced by:  r1filim  35136  r1omhf  35138