Theorem ssiun2sf 29504

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 2806	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴
4		ssiun2sf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
5		nfiu1 4582	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
6	4, 5	nfss 3629	. . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
7	3, 6	nfim 1865	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8		eleq1 2718	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9		ssiun2sf.4	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
10	9	sseq1d 3665	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
11	8, 10	imbi12d 333	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)))
12		ssiun2 4595	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 7, 11, 12	vtoclgf 3295	. 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 1523   ∈ wcel 2030  Ⅎwnfc 2780   ⊆ wss 3607  ∪ ciun 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iun 4554

This theorem is referenced by:  iundisj2f  29529  esum2dlem  30282  voliune  30420  volfiniune  30421