Theorem ttrclselem1 9765

Description: Lemma for ttrclse 9767. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)

Hypothesis

Ref	Expression
ttrclselem.1	⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))

Assertion

Ref	Expression
ttrclselem1	⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑏,𝑤   𝑅,𝑏,𝑤   𝑋,𝑏

Allowed substitution hints:   𝐹(𝑤,𝑏)   𝑁(𝑤,𝑏)   𝑋(𝑤)

Proof of Theorem ttrclselem1

Dummy variables 𝑛 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0suc 7916	. 2 ⊢ (𝑁 ∈ ω → (𝑁 = ∅ ∨ ∃𝑛 ∈ ω 𝑁 = suc 𝑛))
2		ttrclselem.1	. . . . . 6 ⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
3	2	fveq1i 6907	. . . . 5 ⊢ (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4		fveq2 6906	. . . . 5 ⊢ (𝑁 = ∅ → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
5	3, 4	eqtrid 2789	. . . 4 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
6		rdg0g 8467	. . . . . 6 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = Pred(𝑅, 𝐴, 𝑋))
7		predss 6329	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
8	6, 7	eqsstrdi 4028	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
9		rdg0n 8474	. . . . . 6 ⊢ (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = ∅)
10		0ss 4400	. . . . . 6 ⊢ ∅ ⊆ 𝐴
11	9, 10	eqsstrdi 4028	. . . . 5 ⊢ (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
12	8, 11	pm2.61i 182	. . . 4 ⊢ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴
13	5, 12	eqsstrdi 4028	. . 3 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) ⊆ 𝐴)
14		nnon 7893	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
15		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑋)
16		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑏𝑛
17		nfmpt1 5250	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤))
18	17, 15	nfrdg 8454	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
19	2, 18	nfcxfr 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑏𝐹
20	19, 16	nffv 6916	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝐹‘𝑛)
21		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑡)
22	20, 21	nfiun 5023	. . . . . . . . 9 ⊢ Ⅎ𝑏∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡)
23		predeq3 6325	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
24	23	cbviunv 5040	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡)
25		iuneq1 5008	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
26	24, 25	eqtrid 2789	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
27	15, 16, 22, 2, 26	rdgsucmptf 8468	. . . . . . . 8 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
28		iunss 5045	. . . . . . . . 9 ⊢ (∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29		predss 6329	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝐹‘𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
31	28, 30	mprgbir 3068	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
32	27, 31	eqsstrdi 4028	. . . . . . 7 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
33	14, 32	sylan 580	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
34	15, 16, 22, 2, 26	rdgsucmptnf 8469	. . . . . . . 8 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) = ∅)
35	34, 10	eqsstrdi 4028	. . . . . . 7 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) ⊆ 𝐴)
36	35	adantl 481	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
37	33, 36	pm2.61dan 813	. . . . 5 ⊢ (𝑛 ∈ ω → (𝐹‘suc 𝑛) ⊆ 𝐴)
38		fveq2 6906	. . . . . 6 ⊢ (𝑁 = suc 𝑛 → (𝐹‘𝑁) = (𝐹‘suc 𝑛))
39	38	sseq1d 4015	. . . . 5 ⊢ (𝑁 = suc 𝑛 → ((𝐹‘𝑁) ⊆ 𝐴 ↔ (𝐹‘suc 𝑛) ⊆ 𝐴))
40	37, 39	syl5ibrcom 247	. . . 4 ⊢ (𝑛 ∈ ω → (𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴))
41	40	rexlimiv 3148	. . 3 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴)
42	13, 41	jaoi 858	. 2 ⊢ ((𝑁 = ∅ ∨ ∃𝑛 ∈ ω 𝑁 = suc 𝑛) → (𝐹‘𝑁) ⊆ 𝐴)
43	1, 42	syl 17	1 ⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991   ↦ cmpt 5225  Predcpred 6320  Oncon0 6384  suc csuc 6386  ‘cfv 6561  ωcom 7887  reccrdg 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450

This theorem is referenced by:  ttrclselem2  9766