Theorem ssiun2s 4051

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 2392	. 2 ⊢ Ⅎ𝑥𝐶
2		nfcv 2392	. . 3 ⊢ Ⅎ𝑥𝐷
3		nfiu1 4037	. . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfss 3241	. 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
5		ssiun2s.1	. . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
6	5	sseq1d 3277	. 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
7		ssiun2 4050	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8	1, 4, 6, 7	vtoclgaf 2888	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ∪ ciun 4007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  imasaddvallemg  13613