Theorem ttrclse 9656

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 9643	. . . . . . 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 2729	. . . . . . . . . . 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 9655	. . . . . . . . . 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 3155	. . . . . . 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 3448	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	elpred 6279	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥)))
10	9	elv 3449	. . . . . . 7 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
11		resdmss 6196	. . . . . . . . 9 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
12		vex 3448	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
13	8, 12	breldm 5862	. . . . . . . . . 10 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom t++(𝑅 ↾ 𝐴))
14		dmttrcl 9650	. . . . . . . . . 10 ⊢ dom t++(𝑅 ↾ 𝐴) = dom (𝑅 ↾ 𝐴)
15	13, 14	eleqtrdi 2838	. . . . . . . . 9 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom (𝑅 ↾ 𝐴))
16	11, 15	sselid 3941	. . . . . . . 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 8361	. . . . . . 7 ⊢ Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))
20		eluniima 7206	. . . . . . 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 2727	. . . 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 9572	. . . . . . 7 ⊢ ω ∈ V
25	24	funimaex 6588	. . . . . 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 7697	. . . 4 ⊢ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
28	23, 27	eqeltrdi 2836	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
29	28	ralrimiva 3125	. 2 ⊢ (𝑅 Se 𝐴 → ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
30		dfse3 6297	. 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 3044  ∃wrex 3053  Vcvv 3444  ∅c0 4292  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5102   ↦ cmpt 5183   Se wse 5582  dom cdm 5631   ↾ cres 5633   “ cima 5634  Predcpred 6261  suc csuc 6322  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  ωcom 7822  reccrdg 8354  t++cttrcl 9636

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-inf2 9570

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-ttrcl 9637

This theorem is referenced by:  frmin  9678  frr1  9688