Theorem ttrclse 9692

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 9679	. . . . . . 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 2763	. . . . . . . . . . 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 9691	. . . . . . . . . 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 1138	. . . . . . . . 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 463	. . . . . . . 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 3187	. . . . . . 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 286	. . . . . 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 3459	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	elpred 6319	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥)))
10	9	elv 3460	. . . . . . 7 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
11		resdmss 6236	. . . . . . . . 9 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
12		vex 3459	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
13	8, 12	breldm 5898	. . . . . . . . . 10 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom t++(𝑅 ↾ 𝐴))
14		dmttrcl 9686	. . . . . . . . . 10 ⊢ dom t++(𝑅 ↾ 𝐴) = dom (𝑅 ↾ 𝐴)
15	13, 14	eleqtrdi 2873	. . . . . . . . 9 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ dom (𝑅 ↾ 𝐴))
16	11, 15	sselid 3935	. . . . . . . 8 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 → 𝑦 ∈ 𝐴)
17	16	pm4.71ri 569	. . . . . . 7 ⊢ (𝑦t++(𝑅 ↾ 𝐴)𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦t++(𝑅 ↾ 𝐴)𝑥))
18	10, 17	bitr4i 281	. . . . . 6 ⊢ (𝑦 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ↔ 𝑦t++(𝑅 ↾ 𝐴)𝑥)
19		rdgfun 8399	. . . . . . 7 ⊢ Fun rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥))
20		eluniima 7248	. . . . . . 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 317	. . . . 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 2761	. . . 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 9608	. . . . . . 7 ⊢ ω ∈ V
25	24	funimaex 6623	. . . . . 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 7739	. . . 4 ⊢ ∪ (rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑥)) “ ω) ∈ V
28	23, 27	eqeltrdi 2871	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑥 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
29	28	ralrimiva 3157	. 2 ⊢ (𝑅 Se 𝐴 → ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
30		dfse3 6337	. 2 ⊢ (t++(𝑅 ↾ 𝐴) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑥) ∈ V)
31	29, 30	sylibr 237	1 ⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ∅c0 4286  ∪ cuni 4872  ∪ ciun 4956   class class class wbr 5109   ↦ cmpt 5192   Se wse 5612  dom cdm 5661   ↾ cres 5663   “ cima 5664  Predcpred 6301  suc csuc 6362  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ωcom 7858  reccrdg 8392  t++cttrcl 9672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-ttrcl 9673

This theorem is referenced by:  frmin  9717  frr1  9727