Theorem ttrclselem1 9615

Description: Lemma for ttrclse 9617. 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 7824	. 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 6823	. . . . 5 ⊢ (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4		fveq2 6822	. . . . 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 2778	. . . 4 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
6		rdg0g 8346	. . . . . 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 6256	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
8	6, 7	eqsstrdi 3974	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
9		rdg0n 8353	. . . . . 6 ⊢ (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = ∅)
10		0ss 4347	. . . . . 6 ⊢ ∅ ⊆ 𝐴
11	9, 10	eqsstrdi 3974	. . . . 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 3974	. . 3 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) ⊆ 𝐴)
14		nnon 7802	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
15		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑋)
16		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑏𝑛
17		nfmpt1 5188	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤))
18	17, 15	nfrdg 8333	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
19	2, 18	nfcxfr 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑏𝐹
20	19, 16	nffv 6832	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝐹‘𝑛)
21		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑡)
22	20, 21	nfiun 4971	. . . . . . . . 9 ⊢ Ⅎ𝑏∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡)
23		predeq3 6252	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
24	23	cbviunv 4987	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡)
25		iuneq1 4956	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
26	24, 25	eqtrid 2778	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
27	15, 16, 22, 2, 26	rdgsucmptf 8347	. . . . . . . 8 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
28		iunss 4992	. . . . . . . . 9 ⊢ (∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29		predss 6256	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝐹‘𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
31	28, 30	mprgbir 3054	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
32	27, 31	eqsstrdi 3974	. . . . . . 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 8348	. . . . . . . 8 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) = ∅)
35	34, 10	eqsstrdi 3974	. . . . . . 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 812	. . . . 5 ⊢ (𝑛 ∈ ω → (𝐹‘suc 𝑛) ⊆ 𝐴)
38		fveq2 6822	. . . . . 6 ⊢ (𝑁 = suc 𝑛 → (𝐹‘𝑁) = (𝐹‘suc 𝑛))
39	38	sseq1d 3961	. . . . 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 3126	. . 3 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴)
42	13, 41	jaoi 857	. 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 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  ∪ ciun 4939   ↦ cmpt 5170  Predcpred 6247  Oncon0 6306  suc csuc 6308  ‘cfv 6481  ωcom 7796  reccrdg 8328

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329

This theorem is referenced by:  ttrclselem2  9616