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Theorem tc2 8837
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 8833 . . . . 5 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
31, 2ax-mp 5 . . . 4 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
4 trss 4922 . . . . . . 7 (Tr 𝑥 → (𝐴𝑥𝐴𝑥))
54imdistanri 565 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴𝑥 ∧ Tr 𝑥))
65ss2abi 3836 . . . . 5 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 intss 4656 . . . . 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 3797 . . 3 (TC‘𝐴) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
101elintab 4646 . . . . 5 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥))
11 simpl 474 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥)
1210, 11mpgbir 1894 . . . 4 𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
131snss 4472 . . . 4 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13mpbi 221 . . 3 {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
159, 14unssi 3952 . 2 ((TC‘𝐴) ∪ {𝐴}) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
161snid 4368 . . . . 5 𝐴 ∈ {𝐴}
17 elun2 3945 . . . . 5 (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
1816, 17ax-mp 5 . . . 4 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19 uniun 4617 . . . . . . 7 ((TC‘𝐴) ∪ {𝐴}) = ( (TC‘𝐴) ∪ {𝐴})
20 tctr 8835 . . . . . . . . 9 Tr (TC‘𝐴)
21 df-tr 4914 . . . . . . . . 9 (Tr (TC‘𝐴) ↔ (TC‘𝐴) ⊆ (TC‘𝐴))
2220, 21mpbi 221 . . . . . . . 8 (TC‘𝐴) ⊆ (TC‘𝐴)
231unisn 4612 . . . . . . . . 9 {𝐴} = 𝐴
24 tcid 8834 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
251, 24ax-mp 5 . . . . . . . . 9 𝐴 ⊆ (TC‘𝐴)
2623, 25eqsstri 3797 . . . . . . . 8 {𝐴} ⊆ (TC‘𝐴)
2722, 26unssi 3952 . . . . . . 7 ( (TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
2819, 27eqsstri 3797 . . . . . 6 ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29 ssun1 3940 . . . . . 6 (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
3028, 29sstri 3772 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31 df-tr 4914 . . . . 5 (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
3230, 31mpbir 222 . . . 4 Tr ((TC‘𝐴) ∪ {𝐴})
33 fvex 6392 . . . . . 6 (TC‘𝐴) ∈ V
34 snex 5066 . . . . . 6 {𝐴} ∈ V
3533, 34unex 7158 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ∈ V
36 eleq2 2833 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴𝑥𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37 treq 4919 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
3836, 37anbi12d 624 . . . . 5 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
3935, 38elab 3507 . . . 4 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
4018, 32, 39mpbir2an 702 . . 3 ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
41 intss1 4650 . . 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 3779 1 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  Vcvv 3350  cun 3732  wss 3734  {csn 4336   cuni 4596   cint 4635  Tr wtr 4913  cfv 6070  TCctc 8831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-tc 8832
This theorem is referenced by:  tcsni  8838
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