Theorem tcmin 9743

Description: Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)

Assertion

Ref	Expression
tcmin	⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Proof of Theorem tcmin

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

Step	Hyp	Ref	Expression
1		tcvalg 9740	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		fvex 6898	. . . . 5 ⊢ (TC‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2869	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
4		intexab 5306	. . . 4 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	sylibr 237	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6		ssin 4135	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵))
7	6	biimpi 219	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑥 ∩ 𝐵))
8		trin 5223	. . . . . . . 8 ⊢ ((Tr 𝑥 ∧ Tr 𝐵) → Tr (𝑥 ∩ 𝐵))
9	7, 8	anim12i 625	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
10	9	an4s 673	. . . . . 6 ⊢ (((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ∧ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
11	10	expcom 419	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
12		vex 3454	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	inex1 5276	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐵) ∈ V
14		sseq2 3934	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵)))
15		treq 5218	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝑥 ∩ 𝐵)))
16	14, 15	anbi12d 644	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
17	13, 16	elab 3632	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
18		intss1 4922	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
19	17, 18	sylbir 238	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
20		inss2 4134	. . . . . 6 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
21	19, 20	sstrdi 3920	. . . . 5 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)
22	11, 21	syl6 36	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
23	22	exlimdv 1965	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
24	5, 23	syl5com 32	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
25		tcvalg 9740	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
26	25	sseq1d 3939	. 2 ⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) ⊆ 𝐵 ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
27	24, 26	sylibrd 262	1 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∩ cint 4906  Tr wtr 5211  ‘cfv 6538  TCctc 9738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-tc 9739

This theorem is used by:  tcidm  9748  tc0  9749  tcwf  9903  itunitc  10502  grur1  10908