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Theorem brttrcl2 9745
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 9744 . 2 (𝐴t++𝑅𝐵 ↔ ∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2 df-1o 8493 . . . . . . . . 9 1o = suc ∅
32difeq2i 4119 . . . . . . . 8 (ω ∖ 1o) = (ω ∖ suc ∅)
43eleq2i 2821 . . . . . . 7 (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5 peano1 7900 . . . . . . . 8 ∅ ∈ ω
6 eldifsucnn 8691 . . . . . . . 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 4376 . . . . . . . 8 (ω ∖ ∅) = ω
98rexeqi 3322 . . . . . . 7 (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
104, 7, 93bitri 296 . . . . . 6 (𝑚 ∈ (ω ∖ 1o) ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
1110anbi1i 622 . . . . 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 3186 . . . . 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 277 . . . 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 1842 . . 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 3068 . . 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 3283 . . 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 302 . 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 3477 . . . . 5 𝑛 ∈ V
1918sucex 7815 . . . 4 suc 𝑛 ∈ V
20 suceq 6440 . . . . . . 7 (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
2120fneq2d 6653 . . . . . 6 (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚𝑓 Fn suc suc 𝑛))
22 fveqeq2 6911 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝑓𝑚) = 𝐵 ↔ (𝑓‘suc 𝑛) = 𝐵))
2322anbi2d 628 . . . . . 6 (𝑚 = suc 𝑛 → (((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵)))
24 raleq 3320 . . . . . 6 (𝑚 = suc 𝑛 → (∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2521, 23, 243anbi123d 1432 . . . . 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 1916 . . . 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 3525 . . 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 3091 . 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 296 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 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3058  wrex 3067  cdif 3946  c0 4326   class class class wbr 5152  suc csuc 6376   Fn wfn 6548  cfv 6553  ωcom 7876  1oc1o 8486  t++cttrcl 9738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-oadd 8497  df-ttrcl 9739
This theorem is referenced by:  ttrclss  9751  ttrclse  9758
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