Theorem ttrclselem1 9641

Description: Lemma for ttrclse 9643. 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 7838	. 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 6832	. . . . 5 ⊢ (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4		fveq2 6831	. . . . 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 2788	. . . 4 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
6		rdg0g 8360	. . . . . 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 6264	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
8	6, 7	eqsstrdi 3961	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
9		rdg0n 8367	. . . . . 6 ⊢ (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = ∅)
10		0ss 4331	. . . . . 6 ⊢ ∅ ⊆ 𝐴
11	9, 10	eqsstrdi 3961	. . . . 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 183	. . . 4 ⊢ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴
13	5, 12	eqsstrdi 3961	. . 3 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) ⊆ 𝐴)
14		nnon 7816	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
15		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑋)
16		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑏𝑛
17		nfmpt1 5174	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤))
18	17, 15	nfrdg 8347	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
19	2, 18	nfcxfr 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑏𝐹
20	19, 16	nffv 6841	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝐹‘𝑛)
21		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑡)
22	20, 21	nfiun 4956	. . . . . . . . 9 ⊢ Ⅎ𝑏∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡)
23		predeq3 6260	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
24	23	cbviunv 4971	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡)
25		iuneq1 4941	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
26	24, 25	eqtrid 2788	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
27	15, 16, 22, 2, 26	rdgsucmptf 8361	. . . . . . . 8 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
28		iunss 4977	. . . . . . . . 9 ⊢ (∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29		predss 6264	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝐹‘𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
31	28, 30	mprgbir 3062	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
32	27, 31	eqsstrdi 3961	. . . . . . 7 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
33	14, 32	sylan 587	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
34	15, 16, 22, 2, 26	rdgsucmptnf 8362	. . . . . . . 8 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) = ∅)
35	34, 10	eqsstrdi 3961	. . . . . . 7 ⊢ (¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) ⊆ 𝐴)
36	35	adantl 483	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ¬ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
37	33, 36	pm2.61dan 819	. . . . 5 ⊢ (𝑛 ∈ ω → (𝐹‘suc 𝑛) ⊆ 𝐴)
38		fveq2 6831	. . . . . 6 ⊢ (𝑁 = suc 𝑛 → (𝐹‘𝑁) = (𝐹‘suc 𝑛))
39	38	sseq1d 3948	. . . . 5 ⊢ (𝑁 = suc 𝑛 → ((𝐹‘𝑁) ⊆ 𝐴 ↔ (𝐹‘suc 𝑛) ⊆ 𝐴))
40	37, 39	syl5ibrcom 249	. . . 4 ⊢ (𝑛 ∈ ω → (𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴))
41	40	rexlimiv 3135	. . 3 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (𝐹‘𝑁) ⊆ 𝐴)
42	13, 41	jaoi 864	. 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 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  Vcvv 3433   ⊆ wss 3885  ∅c0 4264  ∪ ciun 4924   ↦ cmpt 5156  Predcpred 6255  Oncon0 6314  suc csuc 6316  ‘cfv 6489  ωcom 7810  reccrdg 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343

This theorem is referenced by:  ttrclselem2  9642