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 4005	. . 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 296	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
5	4	ssrdv 3941	1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  ∪ ciun 4948

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3444  df-ss 3920  df-iun 4950

This theorem is referenced by:  iuneq1  4965  iunxdif2  5011  oelim2  8535  fsumiun  15758  ovolfiniun  25475  uniioovol  25553  fusgreghash2wspv  30428  ssdifidllem  33555  esum2dlem  34276  esum2d  34277  carsgclctunlem2  34503  bnj1413  35217  bnj1408  35218  volsupnfl  37945  corclrcl  44092  cotrcltrcl  44110  iuneqfzuzlem  45722  fsumiunss  45964  sge0iunmptlemfi  46800  sge0iunmptlemre  46802  carageniuncllem1  46908  carageniuncllem2  46909  caratheodorylem2  46914  ovnsubaddlem1  46957