Theorem ssiun2s 5053

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 2903	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2903	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 5032	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3988	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 4027	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 5052	. 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 1537   ∈ wcel 2106   ⊆ wss 3963  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-iun 4998

This theorem is referenced by:  fviunfun  7968  onfununi  8380  oaordi  8583  omordi  8603  dffi3  9469  alephordi  10112  domtriomlem  10480  pwxpndom2  10703  wunex2  10776  imasaddvallem  17576  imasvscaval  17585  iundisj2  25598  voliunlem1  25599  volsup  25605  iundisj2fi  32805  constr01  33747  bnj906  34923  bnj1137  34988  bnj1408  35029  cvmliftlem10  35279  cvmliftlem13  35281  sstotbnd2  37761  mapdrvallem3  41629  onsucunifi  43360  fvmptiunrelexplb0d  43674  fvmptiunrelexplb1d  43676  corclrcl  43697  trclrelexplem  43701  corcltrcl  43729  cotrclrcl  43732  iunincfi  45034  iundjiunlem  46415  meaiuninc3v  46440  caratheodorylem1  46482  ovnhoilem1  46557