Theorem ssiun2sf 33019

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 2938	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴
4		ssiun2sf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
5		nfiu1 4990	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
6	4, 5	nfss 3927	. . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
7	3, 6	nfim 1929	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
8		eleq1 2850	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9		ssiun2sf.4	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
10	9	sseq1d 3965	. . . 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 5010	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 7, 11, 12	vtoclgf 3532	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
14	13	pm2.43i 53	1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2909   ⊆ wss 3902  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-v 3455  df-ss 3919  df-iun 4956

This theorem is used by:  iundisj2f  33050  esum2dlem  34589  voliune  34727  volfiniune  34728