Theorem ttrclselem1 9638

Description: Lemma for ttrclse 9640. 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 6836	. . . . 5 ⊢ (𝐹‘𝑁) = (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4		fveq2 6835	. . . . 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 2784	. . . 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 6268	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
8	6, 7	eqsstrdi 3979	. . . . 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 4353	. . . . . 6 ⊢ ∅ ⊆ 𝐴
11	9, 10	eqsstrdi 3979	. . . . 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 3979	. . 3 ⊢ (𝑁 = ∅ → (𝐹‘𝑁) ⊆ 𝐴)
14		nnon 7816	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
15		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑋)
16		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑏𝑛
17		nfmpt1 5198	. . . . . . . . . . . . 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 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑏𝐹
20	19, 16	nffv 6845	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝐹‘𝑛)
21		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑏Pred(𝑅, 𝐴, 𝑡)
22	20, 21	nfiun 4979	. . . . . . . . 9 ⊢ Ⅎ𝑏∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡)
23		predeq3 6264	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
24	23	cbviunv 4995	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤) = ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡)
25		iuneq1 4964	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑛) → ∪ 𝑡 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡))
26	24, 25	eqtrid 2784	. . . . . . . . 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 5001	. . . . . . . . 9 ⊢ (∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29		predss 6268	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝐹‘𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
31	28, 30	mprgbir 3059	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
32	27, 31	eqsstrdi 3979	. . . . . . 7 ⊢ ((𝑛 ∈ On ∧ ∪ 𝑡 ∈ (𝐹‘𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
33	14, 32	sylan 581	. . . . . 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 3979	. . . . . . 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 6835	. . . . . 6 ⊢ (𝑁 = suc 𝑛 → (𝐹‘𝑁) = (𝐹‘suc 𝑛))
39	38	sseq1d 3966	. . . . 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 3131	. . 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 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441   ⊆ wss 3902  ∅c0 4286  ∪ ciun 4947   ↦ cmpt 5180  Predcpred 6259  Oncon0 6318  suc csuc 6320  ‘cfv 6493  ωcom 7810  reccrdg 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  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  9639