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Theorem tc2 9210
 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 9206 . . . . 5 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
31, 2ax-mp 5 . . . 4 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
4 trss 5148 . . . . . . 7 (Tr 𝑥 → (𝐴𝑥𝐴𝑥))
54imdistanri 574 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴𝑥 ∧ Tr 𝑥))
65ss2abi 3972 . . . . 5 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 intss 4860 . . . . 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 3927 . . 3 (TC‘𝐴) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
101elintab 4850 . . . . 5 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥))
11 simpl 487 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥)
1210, 11mpgbir 1802 . . . 4 𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
131snss 4677 . . . 4 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13mpbi 233 . . 3 {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
159, 14unssi 4091 . 2 ((TC‘𝐴) ∪ {𝐴}) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
161snid 4559 . . . . 5 𝐴 ∈ {𝐴}
17 elun2 4083 . . . . 5 (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
1816, 17ax-mp 5 . . . 4 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19 uniun 4824 . . . . . . 7 ((TC‘𝐴) ∪ {𝐴}) = ( (TC‘𝐴) ∪ {𝐴})
20 tctr 9208 . . . . . . . . 9 Tr (TC‘𝐴)
21 df-tr 5140 . . . . . . . . 9 (Tr (TC‘𝐴) ↔ (TC‘𝐴) ⊆ (TC‘𝐴))
2220, 21mpbi 233 . . . . . . . 8 (TC‘𝐴) ⊆ (TC‘𝐴)
231unisn 4821 . . . . . . . . 9 {𝐴} = 𝐴
24 tcid 9207 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
251, 24ax-mp 5 . . . . . . . . 9 𝐴 ⊆ (TC‘𝐴)
2623, 25eqsstri 3927 . . . . . . . 8 {𝐴} ⊆ (TC‘𝐴)
2722, 26unssi 4091 . . . . . . 7 ( (TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
2819, 27eqsstri 3927 . . . . . 6 ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29 ssun1 4078 . . . . . 6 (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
3028, 29sstri 3902 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31 df-tr 5140 . . . . 5 (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
3230, 31mpbir 234 . . . 4 Tr ((TC‘𝐴) ∪ {𝐴})
33 fvex 6672 . . . . . 6 (TC‘𝐴) ∈ V
34 snex 5301 . . . . . 6 {𝐴} ∈ V
3533, 34unex 7468 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ∈ V
36 eleq2 2841 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴𝑥𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37 treq 5145 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
3836, 37anbi12d 634 . . . . 5 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
3935, 38elab 3589 . . . 4 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
4018, 32, 39mpbir2an 711 . . 3 ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
41 intss1 4854 . . 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 3909 1 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  Vcvv 3410   ∪ cun 3857   ⊆ wss 3859  {csn 4523  ∪ cuni 4799  ∩ cint 4839  Tr wtr 5139  ‘cfv 6336  TCctc 9204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460  ax-inf2 9130 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-tc 9205 This theorem is referenced by:  tcsni  9211
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