Theorem ss2iun 4977

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 3943	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2	1	ralimi 3067	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
3		rexim 3071	. . . 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 4962	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6		eliun 4962	. . 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 3955	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ∪ ciun 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-ss 3934  df-iun 4960

This theorem is referenced by:  iuneq2  4978  abnexg  7735  oawordri  8517  omwordri  8539  oewordri  8559  oeworde  8560  r1val1  9746  cfslb2n  10228  imasaddvallem  17499  dprdss  19968  tgcmp  23295  txcmplem1  23535  txcmplem2  23536  xkococnlem  23553  alexsubALT  23945  ptcmplem3  23948  metnrmlem2  24756  uniiccvol  25488  dvfval  25805  gsumpart  33004  bnj1145  34990  bnj1136  34994  filnetlem3  36375  poimirlem32  37653  sstotbnd2  37775  equivtotbnd  37779  trclrelexplem  43707  corcltrcl  43735  cotrclrcl  43738  ovolval5lem2  46658  ovolval5lem3  46659  smflimsuplem7  46831