Metamath Proof Explorer

Theorem tcmin

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)

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

Proof

Step Hyp Ref Expression
1 tcvalg ⊢ ( 𝐴 ∈ 𝑉 → ( TC ‘ 𝐴 ) = ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) } )
2 fvex ⊢ ( TC ‘ 𝐴 ) ∈ V
3 1 2 eqeltrrdi ⊢ ( 𝐴 ∈ 𝑉 → ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) } ∈ V )
4 intexab ⊢ ( ∃ 𝑥 ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) ↔ ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) } ∈ V )
5 3 4 sylibr ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) )
6 ssin ⊢ ( ( 𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵 ) ↔ 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) )
7 6 biimpi ⊢ ( ( 𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) )
8 trin ⊢ ( ( Tr 𝑥 ∧ Tr 𝐵 ) → Tr ( 𝑥 ∩ 𝐵 ) )
9 7 8 anim12i ⊢ ( ( ( 𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵 ) ∧ ( Tr 𝑥 ∧ Tr 𝐵 ) ) → ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) )
10 9 an4s ⊢ ( ( ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) ∧ ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) ) → ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) )
11 10 expcom ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) → ( ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) → ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) ) )
12 vex ⊢ 𝑥 ∈ V
13 12 inex1 ⊢ ( 𝑥 ∩ 𝐵 ) ∈ V
14 sseq2 ⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( 𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ) )
15 treq ⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( Tr 𝑦 ↔ Tr ( 𝑥 ∩ 𝐵 ) ) )
16 14 15 anbi12d ⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) ↔ ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) ) )
17 13 16 elab ⊢ ( ( 𝑥 ∩ 𝐵 ) ∈ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ↔ ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) )
18 intss1 ⊢ ( ( 𝑥 ∩ 𝐵 ) ∈ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ ( 𝑥 ∩ 𝐵 ) )
19 17 18 sylbir ⊢ ( ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ ( 𝑥 ∩ 𝐵 ) )
20 inss2 ⊢ ( 𝑥 ∩ 𝐵 ) ⊆ 𝐵
21 19 20 sstrdi ⊢ ( ( 𝐴 ⊆ ( 𝑥 ∩ 𝐵 ) ∧ Tr ( 𝑥 ∩ 𝐵 ) ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ 𝐵 )
22 11 21 syl6 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) → ( ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ 𝐵 ) )
23 22 exlimdv ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) → ( ∃ 𝑥 ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ 𝐵 ) )
24 5 23 syl5com ⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ 𝐵 ) )
25 tcvalg ⊢ ( 𝐴 ∈ 𝑉 → ( TC ‘ 𝐴 ) = ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } )
26 25 sseq1d ⊢ ( 𝐴 ∈ 𝑉 → ( ( TC ‘ 𝐴 ) ⊆ 𝐵 ↔ ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ Tr 𝑦 ) } ⊆ 𝐵 ) )
27 24 26 sylibrd ⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 ⊆ 𝐵 ∧ Tr 𝐵 ) → ( TC ‘ 𝐴 ) ⊆ 𝐵 ) )