Theorem ss2iun 4953

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun	⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ss2iun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3916	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2	1	ralimi 3075	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
3		rexim 3079	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
4	2, 3	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5		eliun 4938	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6		eliun 4938	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	4, 5, 6	3imtr4g 296	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
8	7	ssrdv 3928	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∪ ciun 4934

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-ral 3053  df-rex 3063  df-v 3432  df-ss 3907  df-iun 4936

This theorem is referenced by:  iuneq2  4954  abnexg  7704  oawordri  8479  omwordri  8501  oewordri  8522  oeworde  8523  r1val1  9704  cfslb2n  10184  imasaddvallem  17487  dprdss  20000  tgcmp  23379  txcmplem1  23619  txcmplem2  23620  xkococnlem  23637  alexsubALT  24029  ptcmplem3  24032  metnrmlem2  24839  uniiccvol  25560  dvfval  25877  gsumpart  33142  bnj1145  35154  bnj1136  35158  tz9.1regs  35297  filnetlem3  36581  poimirlem32  37990  sstotbnd2  38112  equivtotbnd  38116  trclrelexplem  44159  corcltrcl  44187  cotrclrcl  44190  ovolval5lem2  47102  ovolval5lem3  47103  smflimsuplem7  47275