Theorem ttrclse 9687

Description: If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴.

This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)

Assertion

Ref	Expression
ttrclse	⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)

Proof of Theorem ttrclse

Dummy variables 𝑎 𝑏 𝑓 𝑛 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brttrcl2 9674	. . . . . . 7 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑛) = 𝑥) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)))
2		eqid 2730	. . . . . . . . . . 11 ⊢ rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))
3	2	ttrclselem2 9686	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑛) = 𝑥) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
4	3	3expb 1120	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ (𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴)) → (∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑛) = 𝑥) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
5	4	ancoms 458	. . . . . . . 8 ⊢ (((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ ω) → (∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑛) = 𝑥) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
6	5	rexbidva 3156	. . . . . . 7 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑛) = 𝑥) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ ω 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
7	1, 6	bitrid 283	. . . . . 6 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦t++(𝑅 ↾ 𝐴)𝑥 ↔ ∃𝑛 ∈ ω 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
8		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	elpred 6294	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥)))
10	9	elv 3455	. . . . . . 7 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
11		resdmss 6211	. . . . . . . . 9 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
12		vex 3454	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
13	8, 12	breldm 5875	. . . . . . . . . 10 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom t++(𝑅 ↾ 𝐴))
14		dmttrcl 9681	. . . . . . . . . 10 ⊢ dom t++(𝑅 ↾ 𝐴) = dom (𝑅 ↾ 𝐴)
15	13, 14	eleqtrdi 2839	. . . . . . . . 9 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom (𝑅 ↾ 𝐴))
16	11, 15	sselid 3947	. . . . . . . 8 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ 𝐴)
17	16	pm4.71ri 560	. . . . . . 7 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
18	10, 17	bitr4i 278	. . . . . 6 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ 𝑦t++(𝑅 ↾ 𝐴)𝑥)
19		rdgfun 8387	. . . . . . 7 ⊢ Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))
20		eluniima 7227	. . . . . . 7 ⊢ (Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) → (𝑦 ∈ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ↔ ∃𝑛 ∈ ω 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛)))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ↔ ∃𝑛 ∈ ω 𝑦 ∈ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))‘𝑛))
22	7, 18, 21	3bitr4g 314	. . . . 5 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ 𝑦 ∈ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω)))
23	22	eqrdv 2728	. . . 4 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) = ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω))
24		omex 9603	. . . . . . 7 ⊢ ω ∈ V
25	24	funimaex 6608	. . . . . 6 ⊢ (Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) → (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V)
26	19, 25	ax-mp 5	. . . . 5 ⊢ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
27	26	uniex 7720	. . . 4 ⊢ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
28	23, 27	eqeltrdi 2837	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
29	28	ralrimiva 3126	. 2 ⊢ (𝑅 Se 𝐴 → ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
30		dfse3 6312	. 2 ⊢ (t++(𝑅 ↾ 𝐴) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
31	29, 30	sylibr 234	1 ⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450  ∅c0 4299  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191   Se wse 5592  dom cdm 5641   ↾ cres 5643   “ cima 5644  Predcpred 6276  suc csuc 6337  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  ωcom 7845  reccrdg 8380  t++cttrcl 9667

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-ttrcl 9668

This theorem is referenced by:  frmin  9709  frr1  9719