Theorem tctr 9766

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 5233	. . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
3		sseq2 3939	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
4		treq 5222	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
5	3, 4	anbi12d 644	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
6	2, 5	elab 3636	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
7	6	simprbi 503	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
8	1, 7	mprg 3086	. . 3 ⊢ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
9		tcvalg 9764	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
10		treq 5222	. . . 4 ⊢ ((TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → (Tr (TC‘𝐴) ↔ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
11	9, 10	syl 18	. . 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 5228	. . 3 ⊢ Tr ∅
14		fvprc 6900	. . . 4 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
15		treq 5222	. . . 4 ⊢ ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
16	14, 15	syl 18	. . 3 ⊢ (¬ 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
17	13, 16	mpbiri 261	. 2 ⊢ (¬ 𝐴 ∈ V → Tr (TC‘𝐴))
18	12, 17	pm2.61i 184	1 ⊢ Tr (TC‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ∩ cint 4910  Tr wtr 5215  ‘cfv 6558  TCctc 9762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-tc 9763

This theorem is used by:  tc2  9768  tcidm  9772  itunitc1  10525  tcfr  46148