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Theorem tc2 8562
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 8558 . . . . 5 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
31, 2ax-mp 5 . . . 4 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
4 trss 4721 . . . . . . 7 (Tr 𝑥 → (𝐴𝑥𝐴𝑥))
54imdistanri 726 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴𝑥 ∧ Tr 𝑥))
65ss2abi 3653 . . . . 5 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 intss 4463 . . . . 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 3614 . . 3 (TC‘𝐴) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
101elintab 4452 . . . . 5 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥))
11 simpl 473 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐴𝑥)
1210, 11mpgbir 1723 . . . 4 𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
131snss 4286 . . . 4 (𝐴 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13mpbi 220 . . 3 {𝐴} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
159, 14unssi 3766 . 2 ((TC‘𝐴) ∪ {𝐴}) ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
161snid 4179 . . . . 5 𝐴 ∈ {𝐴}
17 elun2 3759 . . . . 5 (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
1816, 17ax-mp 5 . . . 4 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19 uniun 4422 . . . . . . 7 ((TC‘𝐴) ∪ {𝐴}) = ( (TC‘𝐴) ∪ {𝐴})
20 tctr 8560 . . . . . . . . 9 Tr (TC‘𝐴)
21 df-tr 4713 . . . . . . . . 9 (Tr (TC‘𝐴) ↔ (TC‘𝐴) ⊆ (TC‘𝐴))
2220, 21mpbi 220 . . . . . . . 8 (TC‘𝐴) ⊆ (TC‘𝐴)
231unisn 4417 . . . . . . . . 9 {𝐴} = 𝐴
24 tcid 8559 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
251, 24ax-mp 5 . . . . . . . . 9 𝐴 ⊆ (TC‘𝐴)
2623, 25eqsstri 3614 . . . . . . . 8 {𝐴} ⊆ (TC‘𝐴)
2722, 26unssi 3766 . . . . . . 7 ( (TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
2819, 27eqsstri 3614 . . . . . 6 ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29 ssun1 3754 . . . . . 6 (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
3028, 29sstri 3592 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31 df-tr 4713 . . . . 5 (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
3230, 31mpbir 221 . . . 4 Tr ((TC‘𝐴) ∪ {𝐴})
33 fvex 6158 . . . . . 6 (TC‘𝐴) ∈ V
34 snex 4869 . . . . . 6 {𝐴} ∈ V
3533, 34unex 6909 . . . . 5 ((TC‘𝐴) ∪ {𝐴}) ∈ V
36 eleq2 2687 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴𝑥𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37 treq 4718 . . . . . 6 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
3836, 37anbi12d 746 . . . . 5 (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
3935, 38elab 3333 . . . 4 (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
4018, 32, 39mpbir2an 954 . . 3 ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
41 intss1 4457 . . 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 3599 1 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  cun 3553  wss 3555  {csn 4148   cuni 4402   cint 4440  Tr wtr 4712  cfv 5847  TCctc 8556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-tc 8557
This theorem is referenced by:  tcsni  8563
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