Theorem ssiun2sf 30323

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 2969	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴
4		ssiun2sf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
5		nfiu1 4915	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
6	4, 5	nfss 3907	. . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
7	3, 6	nfim 1897	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8		eleq1 2877	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9		ssiun2sf.4	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
10	9	sseq1d 3946	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
11	8, 10	imbi12d 348	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)))
12		ssiun2 4934	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 7, 11, 12	vtoclgf 3513	. 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 1538   ∈ wcel 2111  Ⅎwnfc 2936   ⊆ wss 3881  ∪ ciun 4881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iun 4883

This theorem is referenced by:  iundisj2f  30353  esum2dlem  31461  voliune  31598  volfiniune  31599