Theorem ssiun2sf 30311

Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Hypotheses

Ref	Expression
ssiun2sf.1	⊢ Ⅎ𝑥𝐴
ssiun2sf.2	⊢ Ⅎ𝑥𝐶
ssiun2sf.3	⊢ Ⅎ𝑥𝐷
ssiun2sf.4	⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)

Assertion

Ref	Expression
ssiun2sf	⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem ssiun2sf

Step	Hyp	Ref	Expression
1		ssiun2sf.2	. . 3 ⊢ Ⅎ𝑥𝐶
2		ssiun2sf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	1, 2	nfel 2992	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴
4		ssiun2sf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
5		nfiu1 4953	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
6	4, 5	nfss 3960	. . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
7	3, 6	nfim 1897	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8		eleq1 2900	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9		ssiun2sf.4	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
10	9	sseq1d 3998	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
11	8, 10	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)))
12		ssiun2 4971	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 7, 11, 12	vtoclgf 3565	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
14	13	pm2.43i 52	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2961   ⊆ wss 3936  ∪ ciun 4919

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3943  df-ss 3952  df-iun 4921

This theorem is referenced by:  iundisj2f  30340  esum2dlem  31351  voliune  31488  volfiniune  31489