Theorem ttrclse 9639

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 9626	. . . . . . 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 2739	. . . . . . . . . . 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 9638	. . . . . . . . . 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 1126	. . . . . . . . 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 459	. . . . . . . 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 3161	. . . . . . 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 284	. . . . . 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 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	elpred 6269	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥)))
10	9	elv 3436	. . . . . . 7 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
11		resdmss 6186	. . . . . . . . 9 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
12		vex 3435	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
13	8, 12	breldm 5850	. . . . . . . . . 10 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom t++(𝑅 ↾ 𝐴))
14		dmttrcl 9633	. . . . . . . . . 10 ⊢ dom t++(𝑅 ↾ 𝐴) = dom (𝑅 ↾ 𝐴)
15	13, 14	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom (𝑅 ↾ 𝐴))
16	11, 15	sselid 3913	. . . . . . . 8 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ 𝐴)
17	16	pm4.71ri 565	. . . . . . 7 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
18	10, 17	bitr4i 279	. . . . . 6 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ 𝑦t++(𝑅 ↾ 𝐴)𝑥)
19		rdgfun 8345	. . . . . . 7 ⊢ Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))
20		eluniima 7194	. . . . . . 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 315	. . . . 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 2737	. . . 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 9555	. . . . . . 7 ⊢ ω ∈ V
25	24	funimaex 6573	. . . . . 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 7684	. . . 4 ⊢ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
28	23, 27	eqeltrdi 2847	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
29	28	ralrimiva 3131	. 2 ⊢ (𝑅 Se 𝐴 → ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
30		dfse3 6287	. 2 ⊢ (t++(𝑅 ↾ 𝐴) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
31	29, 30	sylibr 235	1 ⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431  ∅c0 4261  ∪ cuni 4838  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153   Se wse 5569  dom cdm 5618   ↾ cres 5620   “ cima 5621  Predcpred 6251  suc csuc 6312  Fun wfun 6479   Fn wfn 6480  ‘cfv 6485  ωcom 7806  reccrdg 8338  t++cttrcl 9619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-ttrcl 9620

This theorem is referenced by:  frmin  9664  frr1  9674