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Theorem brttrcl2 9754
Description: Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.)
Assertion
Ref Expression
brttrcl2 (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
Distinct variable groups:   𝐴,𝑛,𝑓,𝑎   𝐵,𝑛,𝑓,𝑎   𝑅,𝑛,𝑓,𝑎

Proof of Theorem brttrcl2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 brttrcl 9753 . 2 (𝐴t++𝑅𝐵 ↔ ∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2 df-1o 8506 . . . . . . . . 9 1o = suc ∅
32difeq2i 4123 . . . . . . . 8 (ω ∖ 1o) = (ω ∖ suc ∅)
43eleq2i 2833 . . . . . . 7 (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5 peano1 7910 . . . . . . . 8 ∅ ∈ ω
6 eldifsucnn 8702 . . . . . . . 8 (∅ ∈ ω → (𝑚 ∈ (ω ∖ suc ∅) ↔ ∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛))
75, 6ax-mp 5 . . . . . . 7 (𝑚 ∈ (ω ∖ suc ∅) ↔ ∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛)
8 dif0 4378 . . . . . . . 8 (ω ∖ ∅) = ω
98rexeqi 3325 . . . . . . 7 (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
104, 7, 93bitri 297 . . . . . 6 (𝑚 ∈ (ω ∖ 1o) ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
1110anbi1i 624 . . . . 5 ((𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ (∃𝑛 ∈ ω 𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
12 r19.41v 3189 . . . . 5 (∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ (∃𝑛 ∈ ω 𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
1311, 12bitr4i 278 . . . 4 ((𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
1413exbii 1848 . . 3 (∃𝑚(𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑚𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
15 df-rex 3071 . . 3 (∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑚(𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
16 rexcom4 3288 . . 3 (∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑚𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
1714, 15, 163bitr4i 303 . 2 (∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
18 vex 3484 . . . . 5 𝑛 ∈ V
1918sucex 7826 . . . 4 suc 𝑛 ∈ V
20 suceq 6450 . . . . . . 7 (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
2120fneq2d 6662 . . . . . 6 (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚𝑓 Fn suc suc 𝑛))
22 fveqeq2 6915 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝑓𝑚) = 𝐵 ↔ (𝑓‘suc 𝑛) = 𝐵))
2322anbi2d 630 . . . . . 6 (𝑚 = suc 𝑛 → (((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵)))
24 raleq 3323 . . . . . 6 (𝑚 = suc 𝑛 → (∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2521, 23, 243anbi123d 1438 . . . . 5 (𝑚 = suc 𝑛 → ((𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
2625exbidv 1921 . . . 4 (𝑚 = suc 𝑛 → (∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
2719, 26ceqsexv 3532 . . 3 (∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2827rexbii 3094 . 2 (∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
291, 17, 283bitri 297 1 (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cdif 3948  c0 4333   class class class wbr 5143  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887  1oc1o 8499  t++cttrcl 9747
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-ttrcl 9748
This theorem is referenced by:  ttrclss  9760  ttrclse  9767
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