Theorem ssiun3 32578

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 3979	. 2 ⊢ (𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
2		df-ral 3059	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
3		eliun 4999	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	3	ralbii 3090	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	1, 2, 4	3bitr2ri 300	1 ⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1534   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ⊆ wss 3962  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-ss 3979  df-iun 4997

This theorem is referenced by: (None)