Theorem ssiun2s 5006

Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Hypothesis

Ref	Expression
ssiun2s.1	⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)

Assertion

Ref	Expression
ssiun2s	⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ssiun2s

Step	Hyp	Ref	Expression
1		nfcv 2924	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2924	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 4985	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3929	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 3967	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 5005	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3540	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  ∪ ciun 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-v 3456  df-ss 3921  df-iun 4951

This theorem is referenced by:  fviunfun  7926  onfununi  8312  oaordi  8515  omordi  8535  dffi3  9377  alephordi  10030  domtriomlem  10399  pwxpndom2  10623  wunex2  10696  imasaddvallem  17559  imasvscaval  17568  iundisj2  25611  voliunlem1  25612  volsup  25618  iundisj2fi  32999  constr01  34039  bnj906  35225  bnj1137  35290  bnj1408  35331  cvmliftlem10  35644  cvmliftlem13  35646  ttciunun  36871  sstotbnd2  38273  mapdrvallem3  42270  onsucunifi  43947  fvmptiunrelexplb0d  44260  fvmptiunrelexplb1d  44262  corclrcl  44283  trclrelexplem  44287  corcltrcl  44315  cotrclrcl  44318  iunincfi  45672  iundjiunlem  47033  meaiuninc3v  47058  caratheodorylem1  47100  ovnhoilem1  47175