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 ( 𝐴𝑉 → ( ( 𝐴𝐵 ∧ Tr 𝐵 ) → ( TC ‘ 𝐴 ) ⊆ 𝐵 ) )

Proof

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