Theorem ssiun 4993

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun	⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem ssiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3923	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
2	1	reximi 3070	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
3		r19.37v 3158	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4	2, 3	syl 17	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
5		eliun 4943	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6	4, 5	imbitrrdi 252	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
7	6	ssrdv 3935	1 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3914  df-iun 4941

This theorem is referenced by:  iunss2  4996  iunpwss  5053  iunpw  7704  onfununi  8261  oen0  8501  trcl  9618  rtrclreclem1  14964  rtrclreclem2  14966  constrmon  33757  oacl2g  43433  omcl2  43436  ofoaf  43458  iunlub  48931  iuneqconst2  48933