Theorem ssiun2s 4995

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 2894	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2894	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 4975	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3922	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 3961	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 4994	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 3527	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3914  df-iun 4941

This theorem is referenced by:  fviunfun  7877  onfununi  8261  oaordi  8461  omordi  8481  dffi3  9315  alephordi  9965  domtriomlem  10333  pwxpndom2  10556  wunex2  10629  imasaddvallem  17433  imasvscaval  17442  iundisj2  25477  voliunlem1  25478  volsup  25484  iundisj2fi  32779  constr01  33755  bnj906  34942  bnj1137  35007  bnj1408  35048  cvmliftlem10  35338  cvmliftlem13  35340  sstotbnd2  37824  mapdrvallem3  41755  onsucunifi  43473  fvmptiunrelexplb0d  43787  fvmptiunrelexplb1d  43789  corclrcl  43810  trclrelexplem  43814  corcltrcl  43842  cotrclrcl  43845  iunincfi  45201  iundjiunlem  46567  meaiuninc3v  46592  caratheodorylem1  46634  ovnhoilem1  46709