Metamath Proof Explorer

Theorem ralrnmpo

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015)

Ref Expression
Hypotheses rngop.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
ralrnmpo.2 ⊢ ( 𝑧 = 𝐶 → ( 𝜑 ↔ 𝜓 ) )
Assertion ralrnmpo ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ( ∀ 𝑧 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step Hyp Ref Expression
1 rngop.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2 ralrnmpo.2 ⊢ ( 𝑧 = 𝐶 → ( 𝜑 ↔ 𝜓 ) )
3 1 rnmpo ⊢ ran 𝐹 = { 𝑤 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑤 = 𝐶 }
4 3 raleqi ⊢ ( ∀ 𝑧 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑧 ∈ { 𝑤 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑤 = 𝐶 } 𝜑 )
5 eqeq1 ⊢ ( 𝑤 = 𝑧 → ( 𝑤 = 𝐶 ↔ 𝑧 = 𝐶 ) )
6 5 2rexbidv ⊢ ( 𝑤 = 𝑧 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 ) )
7 6 ralab ⊢ ( ∀ 𝑧 ∈ { 𝑤 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑤 = 𝐶 } 𝜑 ↔ ∀ 𝑧 ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
8 ralcom4 ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑧 ∀ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
9 r19.23v ⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
10 9 albii ⊢ ( ∀ 𝑧 ∀ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
11 8 10 bitr2i ⊢ ( ∀ 𝑧 ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
12 4 7 11 3bitri ⊢ ( ∀ 𝑧 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
13 ralcom4 ⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑧 ∀ 𝑦 ∈ 𝐵 ( 𝑧 = 𝐶 → 𝜑 ) )
14 r19.23v ⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
15 14 albii ⊢ ( ∀ 𝑧 ∀ 𝑦 ∈ 𝐵 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
16 13 15 bitri ⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) )
17 nfv ⊢ Ⅎ 𝑧 𝜓
18 17 2 ceqsalg ⊢ ( 𝐶 ∈ 𝑉 → ( ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ 𝜓 ) )
19 18 ralimi ⊢ ( ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ 𝜓 ) )
20 ralbi ⊢ ( ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ 𝜓 ) → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
21 19 20 syl ⊢ ( ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ( 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
22 16 21 bitr3id ⊢ ( ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ( ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
23 22 ralimi ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
24 ralbi ⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )
25 23 24 syl ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ( ∃ 𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )
26 12 25 syl5bb ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ( ∀ 𝑧 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )