Theorem tcmin 9648

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 9645	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		fvex 6847	. . . . 5 ⊢ (TC‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2845	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
4		intexab 5291	. . . 4 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6		ssin 4191	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵))
7	6	biimpi 216	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑥 ∩ 𝐵))
8		trin 5216	. . . . . . . 8 ⊢ ((Tr 𝑥 ∧ Tr 𝐵) → Tr (𝑥 ∩ 𝐵))
9	7, 8	anim12i 613	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
10	9	an4s 660	. . . . . 6 ⊢ (((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ∧ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
11	10	expcom 413	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
12		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	inex1 5262	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐵) ∈ V
14		sseq2 3960	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵)))
15		treq 5212	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝑥 ∩ 𝐵)))
16	14, 15	anbi12d 632	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
17	13, 16	elab 3634	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
18		intss1 4918	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
19	17, 18	sylbir 235	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
20		inss2 4190	. . . . . 6 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
21	19, 20	sstrdi 3946	. . . . 5 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)
22	11, 21	syl6 35	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
23	22	exlimdv 1934	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
24	5, 23	syl5com 31	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
25		tcvalg 9645	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
26	25	sseq1d 3965	. 2 ⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) ⊆ 𝐵 ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
27	24, 26	sylibrd 259	1 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ∩ cint 4902  Tr wtr 5205  ‘cfv 6492  TCctc 9643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-tc 9644

This theorem is referenced by:  tcidm  9653  tc0  9654  tcwf  9795  itunitc  10331  grur1  10731