Theorem ttrclselem1 9794

Description: Lemma for ttrclse 9796. 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 7934	. 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 6921	. . . . 5 ⊢ (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4		fveq2 6920	. . . . 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 2792	. . . 4 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
6		rdg0g 8483	. . . . . 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 6340	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
8	6, 7	eqsstrdi 4063	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
9		rdg0n 8490	. . . . . 6 ⊢ (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = ∅)
10		0ss 4423	. . . . . 6 ⊢ ∅ ⊆ 𝐴
11	9, 10	eqsstrdi 4063	. . . . 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 4063	. . 3 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) ⊆ 𝐴)
14		nnon 7909	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
15		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑋)
16		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑏𝑛
17		nfmpt1 5274	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤))
18	17, 15	nfrdg 8470	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
19	2, 18	nfcxfr 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑏𝐹
20	19, 16	nffv 6930	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝐹‘𝑛)
21		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑡)
22	20, 21	nfiun 5046	. . . . . . . . 9 ⊢ Ⅎ𝑏∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡)
23		predeq3 6336	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
24	23	cbviunv 5063	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡)
25		iuneq1 5031	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
26	24, 25	eqtrid 2792	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
27	15, 16, 22, 2, 26	rdgsucmptf 8484	. . . . . . . 8 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
28		iunss 5068	. . . . . . . . 9 ⊢ (∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29		predss 6340	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝐹‘𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
31	28, 30	mprgbir 3074	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
32	27, 31	eqsstrdi 4063	. . . . . . 7 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
33	14, 32	sylan 579	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
34	15, 16, 22, 2, 26	rdgsucmptnf 8485	. . . . . . . 8 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) = ∅)
35	34, 10	eqsstrdi 4063	. . . . . . 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 6920	. . . . . 6 ⊢ (𝑁 = suc 𝑛 → (𝐹‘𝑁) = (𝐹‘suc 𝑛))
39	38	sseq1d 4040	. . . . 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 3154	. . 3 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴)
42	13, 41	jaoi 856	. 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 846   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∪ ciun 5015   ↦ cmpt 5249  Predcpred 6331  Oncon0 6395  suc csuc 6397  ‘cfv 6573  ωcom 7903  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  ttrclselem2  9795