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Theorem tctr 9174
Description: Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tctr Tr (TC‘𝐴)

Proof of Theorem tctr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trint 5179 . . . 4 (∀𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 vex 3496 . . . . . 6 𝑦 ∈ V
3 sseq2 3991 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
4 treq 5169 . . . . . . 7 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
53, 4anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑦 ∧ Tr 𝑦)))
62, 5elab 3665 . . . . 5 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴𝑦 ∧ Tr 𝑦))
76simprbi 499 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
81, 7mprg 3150 . . 3 Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
9 tcvalg 9172 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
10 treq 5169 . . . 4 ((TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → (Tr (TC‘𝐴) ↔ Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
119, 10syl 17 . . 3 (𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
128, 11mpbiri 260 . 2 (𝐴 ∈ V → Tr (TC‘𝐴))
13 tr0 5174 . . 3 Tr ∅
14 fvprc 6656 . . . 4 𝐴 ∈ V → (TC‘𝐴) = ∅)
15 treq 5169 . . . 4 ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
1614, 15syl 17 . . 3 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
1713, 16mpbiri 260 . 2 𝐴 ∈ V → Tr (TC‘𝐴))
1812, 17pm2.61i 184 1 Tr (TC‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wa 398   = wceq 1530  wcel 2107  {cab 2797  Vcvv 3493  wss 3934  c0 4289   cint 4867  Tr wtr 5163  cfv 6348  TCctc 9170
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-tc 9171
This theorem is referenced by:  tc2  9176  tcidm  9180  itunitc1  9834
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