Metamath Proof Explorer


Theorem elmptrab2

Description: Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015) (Revised by AV, 26-Mar-2021)

Ref Expression
Hypotheses elmptrab2.f 𝐹 = ( 𝑥 ∈ V ↦ { 𝑦𝐵𝜑 } )
elmptrab2.s1 ( ( 𝑥 = 𝑋𝑦 = 𝑌 ) → ( 𝜑𝜓 ) )
elmptrab2.s2 ( 𝑥 = 𝑋𝐵 = 𝐶 )
elmptrab2.ex 𝐵 ∈ V
elmptrab2.rc ( 𝑌𝐶𝑋𝑊 )
Assertion elmptrab2 ( 𝑌 ∈ ( 𝐹𝑋 ) ↔ ( 𝑌𝐶𝜓 ) )

Proof

Step Hyp Ref Expression
1 elmptrab2.f 𝐹 = ( 𝑥 ∈ V ↦ { 𝑦𝐵𝜑 } )
2 elmptrab2.s1 ( ( 𝑥 = 𝑋𝑦 = 𝑌 ) → ( 𝜑𝜓 ) )
3 elmptrab2.s2 ( 𝑥 = 𝑋𝐵 = 𝐶 )
4 elmptrab2.ex 𝐵 ∈ V
5 elmptrab2.rc ( 𝑌𝐶𝑋𝑊 )
6 4 a1i ( 𝑥 ∈ V → 𝐵 ∈ V )
7 1 2 3 6 elmptrab ( 𝑌 ∈ ( 𝐹𝑋 ) ↔ ( 𝑋 ∈ V ∧ 𝑌𝐶𝜓 ) )
8 3simpc ( ( 𝑋 ∈ V ∧ 𝑌𝐶𝜓 ) → ( 𝑌𝐶𝜓 ) )
9 5 elexd ( 𝑌𝐶𝑋 ∈ V )
10 9 adantr ( ( 𝑌𝐶𝜓 ) → 𝑋 ∈ V )
11 simpl ( ( 𝑌𝐶𝜓 ) → 𝑌𝐶 )
12 simpr ( ( 𝑌𝐶𝜓 ) → 𝜓 )
13 10 11 12 3jca ( ( 𝑌𝐶𝜓 ) → ( 𝑋 ∈ V ∧ 𝑌𝐶𝜓 ) )
14 8 13 impbii ( ( 𝑋 ∈ V ∧ 𝑌𝐶𝜓 ) ↔ ( 𝑌𝐶𝜓 ) )
15 7 14 bitri ( 𝑌 ∈ ( 𝐹𝑋 ) ↔ ( 𝑌𝐶𝜓 ) )