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Theorem brttrcl 9710
Description: Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.)
Assertion
Ref Expression
brttrcl (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
Distinct variable groups:   𝐴,𝑛,𝑓,𝑎   𝐵,𝑛,𝑓,𝑎   𝑅,𝑛,𝑓,𝑎

Proof of Theorem brttrcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relttrcl 9709 . . 3 Rel t++𝑅
21brrelex12i 5730 . 2 (𝐴t++𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 fvex 6903 . . . . . . 7 (𝑓‘∅) ∈ V
4 eleq1 2819 . . . . . . 7 ((𝑓‘∅) = 𝐴 → ((𝑓‘∅) ∈ V ↔ 𝐴 ∈ V))
53, 4mpbii 232 . . . . . 6 ((𝑓‘∅) = 𝐴𝐴 ∈ V)
6 fvex 6903 . . . . . . 7 (𝑓𝑛) ∈ V
7 eleq1 2819 . . . . . . 7 ((𝑓𝑛) = 𝐵 → ((𝑓𝑛) ∈ V ↔ 𝐵 ∈ V))
86, 7mpbii 232 . . . . . 6 ((𝑓𝑛) = 𝐵𝐵 ∈ V)
95, 8anim12i 611 . . . . 5 (((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1093ad2ant2 1132 . . . 4 ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1110exlimiv 1931 . . 3 (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1211rexlimivw 3149 . 2 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13 eqeq2 2742 . . . . . . 7 (𝑥 = 𝐴 → ((𝑓‘∅) = 𝑥 ↔ (𝑓‘∅) = 𝐴))
1413anbi1d 628 . . . . . 6 (𝑥 = 𝐴 → (((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦)))
15143anbi2d 1439 . . . . 5 (𝑥 = 𝐴 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
1615exbidv 1922 . . . 4 (𝑥 = 𝐴 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
1716rexbidv 3176 . . 3 (𝑥 = 𝐴 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
18 eqeq2 2742 . . . . . . 7 (𝑦 = 𝐵 → ((𝑓𝑛) = 𝑦 ↔ (𝑓𝑛) = 𝐵))
1918anbi2d 627 . . . . . 6 (𝑦 = 𝐵 → (((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵)))
20193anbi2d 1439 . . . . 5 (𝑦 = 𝐵 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
2120exbidv 1922 . . . 4 (𝑦 = 𝐵 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
2221rexbidv 3176 . . 3 (𝑦 = 𝐵 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
23 df-ttrcl 9705 . . 3 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
2417, 22, 23brabg 5538 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))))
252, 12, 24pm5.21nii 377 1 (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑛) = 𝐵) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1085   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  cdif 3944  c0 4321   class class class wbr 5147  suc csuc 6365   Fn wfn 6537  cfv 6542  ωcom 7857  1oc1o 8461  t++cttrcl 9704
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-iota 6494  df-fv 6550  df-ttrcl 9705
This theorem is referenced by:  brttrcl2  9711  ssttrcl  9712  ttrcltr  9713
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