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 3932	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2	1	ralimi 3104	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
3		rexim 3108	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
4	2, 3	syl 18	. . 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 299	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
8	7	ssrdv 3944	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∪ ciun 4958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-v 3459  df-ss 3923  df-iun 4960

This theorem is used by:  iuneq2  4978  abnexg  7761  oawordri  8541  omwordri  8563  oewordri  8584  oeworde  8585  r1val1  9765  cfslb2n  10267  imasaddvallem  17605  dprdss  20145  tgcmp  23608  txcmplem1  23849  txcmplem2  23850  xkococnlem  23867  alexsubALT  24259  ptcmplem3  24262  metnrmlem2  25069  uniiccvol  25790  dvfval  26107  gsumpart  33447  bnj1145  35446  bnj1136  35450  tz9.1regs  35604  filnetlem3  36948  poimirlem32  38360  sstotbnd2  38483  equivtotbnd  38487  trclrelexplem  44495  corcltrcl  44523  cotrclrcl  44526  ovolval5lem2  47425  ovolval5lem3  47426  smflimsuplem7  47598