Theorem ssiun2s 4997

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 2891	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2891	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 4977	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3928	. 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 4996	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3531	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  ∪ ciun 4941

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3438  df-ss 3920  df-iun 4943

This theorem is referenced by:  fviunfun  7880  onfununi  8264  oaordi  8464  omordi  8484  dffi3  9321  alephordi  9968  domtriomlem  10336  pwxpndom2  10559  wunex2  10632  imasaddvallem  17433  imasvscaval  17442  iundisj2  25448  voliunlem1  25449  volsup  25455  iundisj2fi  32740  constr01  33709  bnj906  34897  bnj1137  34962  bnj1408  35003  cvmliftlem10  35271  cvmliftlem13  35273  sstotbnd2  37758  mapdrvallem3  41629  onsucunifi  43347  fvmptiunrelexplb0d  43661  fvmptiunrelexplb1d  43663  corclrcl  43684  trclrelexplem  43688  corcltrcl  43716  cotrclrcl  43719  iunincfi  45076  iundjiunlem  46444  meaiuninc3v  46469  caratheodorylem1  46511  ovnhoilem1  46586