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Metamath Proof Explorer

Theorem iunssfOLD

Description: Obsolete version of iunssf as of 2-Feb-2026. (Contributed by Glauco Siliprandi, 3-Mar-2021) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis iunssf.1 ⊢ Ⅎ 𝑥 𝐶
Assertion iunssfOLD ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 iunssf.1 ⊢ Ⅎ 𝑥 𝐶
2 df-iun ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 }
3 2 sseq1i ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 } ⊆ 𝐶 )
4 abss ⊢ ( { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 } ⊆ 𝐶 ↔ ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
5 df-ss ⊢ ( 𝐵 ⊆ 𝐶 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
6 5 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
7 ralcom4 ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
8 1 nfcri ⊢ Ⅎ 𝑥 𝑦 ∈ 𝐶
9 8 r19.23 ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
10 9 albii ⊢ ( ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) ↔ ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
11 6 7 10 3bitrri ⊢ ( ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
12 3 4 11 3bitri ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )