Theorem iunss1 4963

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 4006	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 4952	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 4952	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	1, 2, 3	3imtr4g 298	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
5	4	ssrdv 3942	1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3904  ∪ ciun 4948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3086  df-v 3455  df-ss 3921  df-iun 4950

This theorem is referenced by:  iuneq1  4965  iunxdif2  5010  oelim2  8558  fsumiun  15830  ovolfiniun  25541  uniioovol  25619  fusgreghash2wspv  30481  ssdifidllem  33602  esum2dlem  34348  esum2d  34349  carsgclctunlem2  34575  bnj1413  35292  bnj1408  35293  volsupnfl  38117  corclrcl  44236  cotrcltrcl  44254  iuneqfzuzlem  45863  fsumiunss  46104  sge0iunmptlemfi  46940  sge0iunmptlemre  46942  carageniuncllem1  47048  carageniuncllem2  47049  caratheodorylem2  47054  ovnsubaddlem1  47097