Theorem ssiun2s 5048

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 2905	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2905	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 5027	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3976	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 4015	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 5047	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3576	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∪ ciun 4991

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993

This theorem is referenced by:  fviunfun  7969  onfununi  8381  oaordi  8584  omordi  8604  dffi3  9471  alephordi  10114  domtriomlem  10482  pwxpndom2  10705  wunex2  10778  imasaddvallem  17574  imasvscaval  17583  iundisj2  25584  voliunlem1  25585  volsup  25591  iundisj2fi  32799  constr01  33783  bnj906  34944  bnj1137  35009  bnj1408  35050  cvmliftlem10  35299  cvmliftlem13  35301  sstotbnd2  37781  mapdrvallem3  41648  onsucunifi  43383  fvmptiunrelexplb0d  43697  fvmptiunrelexplb1d  43699  corclrcl  43720  trclrelexplem  43724  corcltrcl  43752  cotrclrcl  43755  iunincfi  45099  iundjiunlem  46474  meaiuninc3v  46499  caratheodorylem1  46541  ovnhoilem1  46616