Theorem ssiun3 31655

Description: Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)

Assertion

Ref	Expression
ssiun3	⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ssiun3

Step	Hyp	Ref	Expression
1		dfss2 3964	. 2 ⊢ (𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
2		df-ral 3061	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
3		eliun 4994	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	3	ralbii 3092	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	1, 2, 4	3bitr2ri 299	1 ⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ⊆ wss 3944  ∪ ciun 4990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-in 3951  df-ss 3961  df-iun 4992

This theorem is referenced by: (None)