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Theorem tc2 9173
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 9169 . . . . 5 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
31, 2ax-mp 5 . . . 4 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
4 trss 5178 . . . . . . 7 (Tr 𝑥 → (𝐴𝑥𝐴𝑥))
54imdistanri 570 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴𝑥 ∧ Tr 𝑥))
65ss2abi 4047 . . . . 5 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 intss 4895 . . . . 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 4005 . . 3 (TC‘𝐴) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
101elintab 4885 . . . . 5 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥))
11 simpl 483 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥)
1210, 11mpgbir 1793 . . . 4 𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
131snss 4717 . . . 4 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13mpbi 231 . . 3 {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
159, 14unssi 4165 . 2 ((TC‘𝐴) ∪ {𝐴}) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
161snid 4598 . . . . 5 𝐴 ∈ {𝐴}
17 elun2 4157 . . . . 5 (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
1816, 17ax-mp 5 . . . 4 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19 uniun 4856 . . . . . . 7 ((TC‘𝐴) ∪ {𝐴}) = ( (TC‘𝐴) ∪ {𝐴})
20 tctr 9171 . . . . . . . . 9 Tr (TC‘𝐴)
21 df-tr 5170 . . . . . . . . 9 (Tr (TC‘𝐴) ↔ (TC‘𝐴) ⊆ (TC‘𝐴))
2220, 21mpbi 231 . . . . . . . 8 (TC‘𝐴) ⊆ (TC‘𝐴)
231unisn 4853 . . . . . . . . 9 {𝐴} = 𝐴
24 tcid 9170 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
251, 24ax-mp 5 . . . . . . . . 9 𝐴 ⊆ (TC‘𝐴)
2623, 25eqsstri 4005 . . . . . . . 8 {𝐴} ⊆ (TC‘𝐴)
2722, 26unssi 4165 . . . . . . 7 ( (TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
2819, 27eqsstri 4005 . . . . . 6 ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29 ssun1 4152 . . . . . 6 (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
3028, 29sstri 3980 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31 df-tr 5170 . . . . 5 (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
3230, 31mpbir 232 . . . 4 Tr ((TC‘𝐴) ∪ {𝐴})
33 fvex 6680 . . . . . 6 (TC‘𝐴) ∈ V
34 snex 5328 . . . . . 6 {𝐴} ∈ V
3533, 34unex 7458 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ∈ V
36 eleq2 2906 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴𝑥𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37 treq 5175 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
3836, 37anbi12d 630 . . . . 5 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
3935, 38elab 3671 . . . 4 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
4018, 32, 39mpbir2an 707 . . 3 ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
41 intss1 4889 . . 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 3987 1 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2804  Vcvv 3500  cun 3938  wss 3940  {csn 4564   cuni 4837   cint 4874  Tr wtr 5169  cfv 6352  TCctc 9167
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-tc 9168
This theorem is referenced by:  tcsni  9174
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