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Theorem tc2 8782
Description: A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1 𝐴 ∈ V
Assertion
Ref Expression
tc2 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
Distinct variable group:   𝑥,𝐴

Proof of Theorem tc2
StepHypRef Expression
1 tc2.1 . . . . 5 𝐴 ∈ V
2 tcvalg 8778 . . . . 5 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
31, 2ax-mp 5 . . . 4 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
4 trss 4895 . . . . . . 7 (Tr 𝑥 → (𝐴𝑥𝐴𝑥))
54imdistanri 559 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴𝑥 ∧ Tr 𝑥))
65ss2abi 3823 . . . . 5 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 intss 4632 . . . . 5 ({𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
86, 7ax-mp 5 . . . 4 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
93, 8eqsstri 3784 . . 3 (TC‘𝐴) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
101elintab 4622 . . . . 5 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥))
11 simpl 468 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥)
1210, 11mpgbir 1874 . . . 4 𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
131snss 4451 . . . 4 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13mpbi 220 . . 3 {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
159, 14unssi 3939 . 2 ((TC‘𝐴) ∪ {𝐴}) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
161snid 4347 . . . . 5 𝐴 ∈ {𝐴}
17 elun2 3932 . . . . 5 (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
1816, 17ax-mp 5 . . . 4 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19 uniun 4593 . . . . . . 7 ((TC‘𝐴) ∪ {𝐴}) = ( (TC‘𝐴) ∪ {𝐴})
20 tctr 8780 . . . . . . . . 9 Tr (TC‘𝐴)
21 df-tr 4887 . . . . . . . . 9 (Tr (TC‘𝐴) ↔ (TC‘𝐴) ⊆ (TC‘𝐴))
2220, 21mpbi 220 . . . . . . . 8 (TC‘𝐴) ⊆ (TC‘𝐴)
231unisn 4589 . . . . . . . . 9 {𝐴} = 𝐴
24 tcid 8779 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
251, 24ax-mp 5 . . . . . . . . 9 𝐴 ⊆ (TC‘𝐴)
2623, 25eqsstri 3784 . . . . . . . 8 {𝐴} ⊆ (TC‘𝐴)
2722, 26unssi 3939 . . . . . . 7 ( (TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
2819, 27eqsstri 3784 . . . . . 6 ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29 ssun1 3927 . . . . . 6 (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
3028, 29sstri 3761 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31 df-tr 4887 . . . . 5 (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
3230, 31mpbir 221 . . . 4 Tr ((TC‘𝐴) ∪ {𝐴})
33 fvex 6342 . . . . . 6 (TC‘𝐴) ∈ V
34 snex 5036 . . . . . 6 {𝐴} ∈ V
3533, 34unex 7103 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ∈ V
36 eleq2 2839 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴𝑥𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37 treq 4892 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
3836, 37anbi12d 616 . . . . 5 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
3935, 38elab 3501 . . . 4 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
4018, 32, 39mpbir2an 690 . . 3 ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
41 intss1 4626 . . 3 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ ((TC‘𝐴) ∪ {𝐴}))
4240, 41ax-mp 5 . 2 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ ((TC‘𝐴) ∪ {𝐴})
4315, 42eqssi 3768 1 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  cun 3721  wss 3723  {csn 4316   cuni 4574   cint 4611  Tr wtr 4886  cfv 6031  TCctc 8776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-tc 8777
This theorem is referenced by:  tcsni  8783
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