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Metamath Proof Explorer

Theorem disjrdx

Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017)

Ref Expression
Hypotheses disjrdx.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐶 )
disjrdx.2 ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → 𝐷 = 𝐵 )
Assertion disjrdx ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷 ) )

Proof

Step Hyp Ref Expression
1 disjrdx.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐶 )
2 disjrdx.2 ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → 𝐷 = 𝐵 )
3 f1of ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 → 𝐹 : 𝐴 ⟶ 𝐶 )
4 1 3 syl ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )
5 4 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐶 )
6 f1ofveu ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ∧ 𝑦 ∈ 𝐶 ) → ∃! 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 )
7 1 6 sylan ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃! 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 )
8 eqcom ⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ 𝑦 = ( 𝐹 ‘ 𝑥 ) )
9 8 reubii ⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ ∃! 𝑥 ∈ 𝐴 𝑦 = ( 𝐹 ‘ 𝑥 ) )
10 7 9 sylib ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃! 𝑥 ∈ 𝐴 𝑦 = ( 𝐹 ‘ 𝑥 ) )
11 2 eleq2d ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → ( 𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵 ) )
12 5 10 11 rmoxfrd ⊢ ( 𝜑 → ( ∃* 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ) )
13 12 bicomd ⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃* 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ) )
14 13 albidv ⊢ ( 𝜑 → ( ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ) )
15 df-disj ⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 )
16 df-disj ⊢ ( Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 )
17 14 15 16 3bitr4g ⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷 ) )