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Theorem brttrcl2 9708
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 9707 . 2 (𝐴t++𝑅𝐵 ↔ ∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓𝑚) = 𝐵) ∧ ∀𝑎𝑚 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
2 df-1o 8464 . . . . . . . . 9 1o = suc ∅
32difeq2i 4114 . . . . . . . 8 (ω ∖ 1o) = (ω ∖ suc ∅)
43eleq2i 2819 . . . . . . 7 (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5 peano1 7875 . . . . . . . 8 ∅ ∈ ω
6 eldifsucnn 8662 . . . . . . . 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 4367 . . . . . . . 8 (ω ∖ ∅) = ω
98rexeqi 3318 . . . . . . 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 623 . . . . 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 3182 . . . . 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 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 3065 . . 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 3279 . . 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 3472 . . . . 5 𝑛 ∈ V
1918sucex 7790 . . . 4 suc 𝑛 ∈ V
20 suceq 6423 . . . . . . 7 (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
2120fneq2d 6636 . . . . . 6 (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚𝑓 Fn suc suc 𝑛))
22 fveqeq2 6893 . . . . . . 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 3316 . . . . . 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 3520 . . 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 3088 . 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 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3055  wrex 3064  cdif 3940  c0 4317   class class class wbr 5141  suc csuc 6359   Fn wfn 6531  cfv 6536  ωcom 7851  1oc1o 8457  t++cttrcl 9701
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-oadd 8468  df-ttrcl 9702
This theorem is referenced by:  ttrclss  9714  ttrclse  9721
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