Theorem ssiun3 32428

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		df-ss 3961	. 2 ⊢ (𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
2		df-ral 3051	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
3		eliun 5001	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	3	ralbii 3082	. 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 1531   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ⊆ wss 3944  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-v 3463  df-ss 3961  df-iun 4999

This theorem is referenced by: (None)