Metamath Proof Explorer


Theorem dfiin2g

Description: Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009)

Ref Expression
Assertion dfiin2g ( ∀ 𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } )

Proof

Step Hyp Ref Expression
1 df-ral ( ∀ 𝑥𝐴 𝑤𝐵 ↔ ∀ 𝑥 ( 𝑥𝐴𝑤𝐵 ) )
2 df-ral ( ∀ 𝑥𝐴 𝐵𝐶 ↔ ∀ 𝑥 ( 𝑥𝐴𝐵𝐶 ) )
3 eleq2 ( 𝑧 = 𝐵 → ( 𝑤𝑧𝑤𝐵 ) )
4 3 biimprcd ( 𝑤𝐵 → ( 𝑧 = 𝐵𝑤𝑧 ) )
5 4 alrimiv ( 𝑤𝐵 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) )
6 eqid 𝐵 = 𝐵
7 eqeq1 ( 𝑧 = 𝐵 → ( 𝑧 = 𝐵𝐵 = 𝐵 ) )
8 7 3 imbi12d ( 𝑧 = 𝐵 → ( ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ( 𝐵 = 𝐵𝑤𝐵 ) ) )
9 8 spcgv ( 𝐵𝐶 → ( ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) → ( 𝐵 = 𝐵𝑤𝐵 ) ) )
10 6 9 mpii ( 𝐵𝐶 → ( ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) → 𝑤𝐵 ) )
11 5 10 impbid2 ( 𝐵𝐶 → ( 𝑤𝐵 ↔ ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) )
12 11 imim2i ( ( 𝑥𝐴𝐵𝐶 ) → ( 𝑥𝐴 → ( 𝑤𝐵 ↔ ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
13 12 pm5.74d ( ( 𝑥𝐴𝐵𝐶 ) → ( ( 𝑥𝐴𝑤𝐵 ) ↔ ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
14 13 alimi ( ∀ 𝑥 ( 𝑥𝐴𝐵𝐶 ) → ∀ 𝑥 ( ( 𝑥𝐴𝑤𝐵 ) ↔ ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
15 albi ( ∀ 𝑥 ( ( 𝑥𝐴𝑤𝐵 ) ↔ ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) → ( ∀ 𝑥 ( 𝑥𝐴𝑤𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
16 14 15 syl ( ∀ 𝑥 ( 𝑥𝐴𝐵𝐶 ) → ( ∀ 𝑥 ( 𝑥𝐴𝑤𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
17 2 16 sylbi ( ∀ 𝑥𝐴 𝐵𝐶 → ( ∀ 𝑥 ( 𝑥𝐴𝑤𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ) )
18 df-ral ( ∀ 𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) )
19 18 albii ( ∀ 𝑧𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ∀ 𝑧𝑥 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) )
20 alcom ( ∀ 𝑥𝑧 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) ↔ ∀ 𝑧𝑥 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) )
21 19 20 bitr4i ( ∀ 𝑧𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ∀ 𝑥𝑧 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) )
22 r19.23v ( ∀ 𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ( ∃ 𝑥𝐴 𝑧 = 𝐵𝑤𝑧 ) )
23 vex 𝑧 ∈ V
24 eqeq1 ( 𝑦 = 𝑧 → ( 𝑦 = 𝐵𝑧 = 𝐵 ) )
25 24 rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑥𝐴 𝑦 = 𝐵 ↔ ∃ 𝑥𝐴 𝑧 = 𝐵 ) )
26 23 25 elab ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } ↔ ∃ 𝑥𝐴 𝑧 = 𝐵 )
27 26 imbi1i ( ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) ↔ ( ∃ 𝑥𝐴 𝑧 = 𝐵𝑤𝑧 ) )
28 22 27 bitr4i ( ∀ 𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) )
29 28 albii ( ∀ 𝑧𝑥𝐴 ( 𝑧 = 𝐵𝑤𝑧 ) ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) )
30 19.21v ( ∀ 𝑧 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) ↔ ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) )
31 30 albii ( ∀ 𝑥𝑧 ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) )
32 21 29 31 3bitr3ri ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑧 ( 𝑧 = 𝐵𝑤𝑧 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) )
33 17 32 bitrdi ( ∀ 𝑥𝐴 𝐵𝐶 → ( ∀ 𝑥 ( 𝑥𝐴𝑤𝐵 ) ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) ) )
34 1 33 bitrid ( ∀ 𝑥𝐴 𝐵𝐶 → ( ∀ 𝑥𝐴 𝑤𝐵 ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) ) )
35 34 abbidv ( ∀ 𝑥𝐴 𝐵𝐶 → { 𝑤 ∣ ∀ 𝑥𝐴 𝑤𝐵 } = { 𝑤 ∣ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) } )
36 df-iin 𝑥𝐴 𝐵 = { 𝑤 ∣ ∀ 𝑥𝐴 𝑤𝐵 }
37 df-int { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } = { 𝑤 ∣ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } → 𝑤𝑧 ) }
38 35 36 37 3eqtr4g ( ∀ 𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } )