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Theorem brttrcl2 9752
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 9751 . 2 (𝐴t++𝑅𝐵 ↔ ∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2 df-1o 8505 . . . . . . . . 9 1o = suc ∅
32difeq2i 4133 . . . . . . . 8 (ω ∖ 1o) = (ω ∖ suc ∅)
43eleq2i 2831 . . . . . . 7 (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5 peano1 7911 . . . . . . . 8 ∅ ∈ ω
6 eldifsucnn 8701 . . . . . . . 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 4384 . . . . . . . 8 (ω ∖ ∅) = ω
98rexeqi 3323 . . . . . . 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 3187 . . . . 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 1845 . . 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 3069 . . 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 3286 . . 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 3482 . . . . 5 𝑛 ∈ V
1918sucex 7826 . . . 4 suc 𝑛 ∈ V
20 suceq 6452 . . . . . . 7 (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
2120fneq2d 6663 . . . . . 6 (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚𝑓 Fn suc suc 𝑛))
22 fveqeq2 6916 . . . . . . 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 3321 . . . . . 6 (𝑚 = suc 𝑛 → (∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2521, 23, 243anbi123d 1435 . . . . 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 1919 . . . 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 3530 . . 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 3092 . 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 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  cdif 3960  c0 4339   class class class wbr 5148  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  1oc1o 8498  t++cttrcl 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-ttrcl 9746
This theorem is referenced by:  ttrclss  9758  ttrclse  9765
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