Theorem ssiun2s 5024

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 2898	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2898	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 5003	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3951	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 3990	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 5023	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3555	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ∪ ciun 4967

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 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3461  df-ss 3943  df-iun 4969

This theorem is referenced by:  fviunfun  7943  onfununi  8355  oaordi  8558  omordi  8578  dffi3  9443  alephordi  10088  domtriomlem  10456  pwxpndom2  10679  wunex2  10752  imasaddvallem  17543  imasvscaval  17552  iundisj2  25502  voliunlem1  25503  volsup  25509  iundisj2fi  32774  constr01  33776  bnj906  34961  bnj1137  35026  bnj1408  35067  cvmliftlem10  35316  cvmliftlem13  35318  sstotbnd2  37798  mapdrvallem3  41665  onsucunifi  43394  fvmptiunrelexplb0d  43708  fvmptiunrelexplb1d  43710  corclrcl  43731  trclrelexplem  43735  corcltrcl  43763  cotrclrcl  43766  iunincfi  45118  iundjiunlem  46488  meaiuninc3v  46513  caratheodorylem1  46555  ovnhoilem1  46630