Theorem iunss1 5011

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss1	⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iunss1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 4051	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 5001	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 5001	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	1, 2, 3	3imtr4g 295	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
5	4	ssrdv 3988	1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃wrex 3070   ⊆ wss 3948  ∪ ciun 4997

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 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-iun 4999

This theorem is referenced by:  iuneq1  5013  iunxdif2  5056  oelim2  8597  fsumiun  15771  ovolfiniun  25242  uniioovol  25320  fusgreghash2wspv  29843  esum2dlem  33376  esum2d  33377  carsgclctunlem2  33604  bnj1413  34332  bnj1408  34333  volsupnfl  36836  corclrcl  42760  cotrcltrcl  42778  iuneqfzuzlem  44343  fsumiunss  44590  sge0iunmptlemfi  45428  sge0iunmptlemre  45430  carageniuncllem1  45536  carageniuncllem2  45537  caratheodorylem2  45542  ovnsubaddlem1  45585