Theorem ttrclse 9648

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 9635	. . . . . . 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 2737	. . . . . . . . . . 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 9647	. . . . . . . . . 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 1121	. . . . . . . . 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 3160	. . . . . . 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 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	elpred 6284	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥)))
10	9	elv 3447	. . . . . . 7 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
11		resdmss 6201	. . . . . . . . 9 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
12		vex 3446	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
13	8, 12	breldm 5865	. . . . . . . . . 10 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom t++(𝑅 ↾ 𝐴))
14		dmttrcl 9642	. . . . . . . . . 10 ⊢ dom t++(𝑅 ↾ 𝐴) = dom (𝑅 ↾ 𝐴)
15	13, 14	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom (𝑅 ↾ 𝐴))
16	11, 15	sselid 3933	. . . . . . . 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 8357	. . . . . . 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 2735	. . . 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 9564	. . . . . . 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 7696	. . . 4 ⊢ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
28	23, 27	eqeltrdi 2845	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
29	28	ralrimiva 3130	. 2 ⊢ (𝑅 Se 𝐴 → ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
30		dfse3 6302	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442  ∅c0 4287  ∪ cuni 4865  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181   Se wse 5583  dom cdm 5632   ↾ cres 5634   “ cima 5635  Predcpred 6266  suc csuc 6327  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500  ωcom 7818  reccrdg 8350  t++cttrcl 9628

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 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-inf2 9562

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 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-ttrcl 9629

This theorem is referenced by:  frmin  9673  frr1  9683