Metamath Proof Explorer


Theorem elintima

Description: Element of intersection of images. (Contributed by RP, 13-Apr-2020)

Ref Expression
Assertion elintima ( 𝑦 { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } ↔ ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )

Proof

Step Hyp Ref Expression
1 vex 𝑦 ∈ V
2 1 elint2 ( 𝑦 { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } ↔ ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } 𝑦𝑧 )
3 elequ2 ( 𝑧 = 𝑥 → ( 𝑦𝑧𝑦𝑥 ) )
4 3 ralab2 ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } 𝑦𝑧 ↔ ∀ 𝑥 ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) → 𝑦𝑥 ) )
5 df-rex ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) ↔ ∃ 𝑎 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) )
6 5 imbi1i ( ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) → 𝑦𝑥 ) ↔ ( ∃ 𝑎 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) )
7 19.23v ( ∀ 𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) ↔ ( ∃ 𝑎 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) )
8 simpr ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑥 = ( 𝑎𝐵 ) )
9 8 eleq2d ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ( 𝑦𝑥𝑦 ∈ ( 𝑎𝐵 ) ) )
10 9 pm5.74i ( ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) ↔ ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦 ∈ ( 𝑎𝐵 ) ) )
11 1 elima ( 𝑦 ∈ ( 𝑎𝐵 ) ↔ ∃ 𝑏𝐵 𝑏 𝑎 𝑦 )
12 df-br ( 𝑏 𝑎 𝑦 ↔ ⟨ 𝑏 , 𝑦 ⟩ ∈ 𝑎 )
13 12 rexbii ( ∃ 𝑏𝐵 𝑏 𝑎 𝑦 ↔ ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )
14 11 13 bitri ( 𝑦 ∈ ( 𝑎𝐵 ) ↔ ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )
15 14 imbi2i ( ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦 ∈ ( 𝑎𝐵 ) ) ↔ ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
16 10 15 bitri ( ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) ↔ ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
17 16 albii ( ∀ 𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → 𝑦𝑥 ) ↔ ∀ 𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
18 6 7 17 3bitr2i ( ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) → 𝑦𝑥 ) ↔ ∀ 𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
19 18 albii ( ∀ 𝑥 ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) → 𝑦𝑥 ) ↔ ∀ 𝑥𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
20 19.23v ( ∀ 𝑥 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ( ∃ 𝑥 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
21 vex 𝑎 ∈ V
22 21 imaex ( 𝑎𝐵 ) ∈ V
23 22 isseti 𝑥 𝑥 = ( 𝑎𝐵 )
24 19.42v ( ∃ 𝑥 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) ↔ ( 𝑎𝐴 ∧ ∃ 𝑥 𝑥 = ( 𝑎𝐵 ) ) )
25 23 24 mpbiran2 ( ∃ 𝑥 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) ↔ 𝑎𝐴 )
26 25 imbi1i ( ( ∃ 𝑥 ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ( 𝑎𝐴 → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
27 20 26 bitri ( ∀ 𝑥 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ( 𝑎𝐴 → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
28 27 albii ( ∀ 𝑎𝑥 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ∀ 𝑎 ( 𝑎𝐴 → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
29 alcom ( ∀ 𝑥𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ∀ 𝑎𝑥 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
30 df-ral ( ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ↔ ∀ 𝑎 ( 𝑎𝐴 → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) )
31 28 29 30 3bitr4i ( ∀ 𝑥𝑎 ( ( 𝑎𝐴𝑥 = ( 𝑎𝐵 ) ) → ∃ 𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 ) ↔ ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )
32 19 31 bitri ( ∀ 𝑥 ( ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) → 𝑦𝑥 ) ↔ ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )
33 4 32 bitri ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } 𝑦𝑧 ↔ ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )
34 2 33 bitri ( 𝑦 { 𝑥 ∣ ∃ 𝑎𝐴 𝑥 = ( 𝑎𝐵 ) } ↔ ∀ 𝑎𝐴𝑏𝐵𝑏 , 𝑦 ⟩ ∈ 𝑎 )