Theorem tctr 9655

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 5219	. . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		vex 3442	. . . . . 6 ⊢ 𝑦 ∈ V
3		sseq2 3964	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
4		treq 5209	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
5	3, 4	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
6	2, 5	elab 3637	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
7	6	simprbi 496	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
8	1, 7	mprg 3050	. . 3 ⊢ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
9		tcvalg 9653	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
10		treq 5209	. . . 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 258	. 2 ⊢ (𝐴 ∈ V → Tr (TC‘𝐴))
13		tr0 5214	. . 3 ⊢ Tr ∅
14		fvprc 6818	. . . 4 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
15		treq 5209	. . . 4 ⊢ ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
16	14, 15	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
17	13, 16	mpbiri 258	. 2 ⊢ (¬ 𝐴 ∈ V → Tr (TC‘𝐴))
18	12, 17	pm2.61i 182	1 ⊢ Tr (TC‘𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3438   ⊆ wss 3905  ∅c0 4286  ∩ cint 4899  Tr wtr 5202  ‘cfv 6486  TCctc 9651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-tc 9652

This theorem is referenced by:  tc2  9657  tcidm  9661  itunitc1  10333  tcfr  44937