Theorem tctr 9166

Description: Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)

Assertion

Ref	Expression
tctr	⊢ Tr (TC‘𝐴)

Proof of Theorem tctr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trint 5152	. . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
3		sseq2 3941	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
4		treq 5142	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
5	3, 4	anbi12d 633	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
6	2, 5	elab 3615	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
7	6	simprbi 500	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
8	1, 7	mprg 3120	. . 3 ⊢ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
9		tcvalg 9164	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
10		treq 5142	. . . 4 ⊢ ((TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → (Tr (TC‘𝐴) ↔ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
11	9, 10	syl 17	. . 3 ⊢ (𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
12	8, 11	mpbiri 261	. 2 ⊢ (𝐴 ∈ V → Tr (TC‘𝐴))
13		tr0 5147	. . 3 ⊢ Tr ∅
14		fvprc 6638	. . . 4 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
15		treq 5142	. . . 4 ⊢ ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
16	14, 15	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
17	13, 16	mpbiri 261	. 2 ⊢ (¬ 𝐴 ∈ V → Tr (TC‘𝐴))
18	12, 17	pm2.61i 185	1 ⊢ Tr (TC‘𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838  Tr wtr 5136  ‘cfv 6324  TCctc 9162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-tc 9163

This theorem is referenced by:  tc2  9168  tcidm  9172  itunitc1  9831