Theorem ssiun2s 5052

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 2904	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2904	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 5032	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3975	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 4014	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 5051	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3565	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ⊆ wss 3949  ∪ ciun 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-iun 5000

This theorem is referenced by:  fviunfun  7931  onfununi  8341  oaordi  8546  omordi  8566  dffi3  9426  alephordi  10069  domtriomlem  10437  pwxpndom2  10660  wunex2  10733  imasaddvallem  17475  imasvscaval  17484  iundisj2  25066  voliunlem1  25067  volsup  25073  iundisj2fi  32039  bnj906  33972  bnj1137  34037  bnj1408  34078  cvmliftlem10  34316  cvmliftlem13  34318  sstotbnd2  36690  mapdrvallem3  40565  onsucunifi  42168  fvmptiunrelexplb0d  42483  fvmptiunrelexplb1d  42485  corclrcl  42506  trclrelexplem  42510  corcltrcl  42538  cotrclrcl  42541  iunincfi  43831  iundjiunlem  45223  meaiuninc3v  45248  caratheodorylem1  45290  ovnhoilem1  45365