Theorem ssiun2s 2020

Description: Subset relationship for an indexed union.

Hypothesis

Ref	Expression
ssiun2s.1	⊢ (x = C → B = D)

Assertion

Ref	Expression
ssiun2s	⊢ (C ∈ A → D ⊆ ∪x ∈ A B)

Distinct variable group(s):   x,A   x,C   x,D

Proof of Theorem ssiun2s

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (y ∈ C → ∀x y ∈ C)
2		ax-17 925	. . . 4 ⊢ (C ∈ A → ∀x C ∈ A)
3		ax-17 925	. . . . 5 ⊢ (y ∈ D → ∀x y ∈ D)
4		hbiu1 2012	. . . . 5 ⊢ (y ∈ ∪x ∈ A B → ∀x y ∈ ∪x ∈ A B)
5	3, 4	hbss 1501	. . . 4 ⊢ (D ⊆ ∪x ∈ A B → ∀x D ⊆ ∪x ∈ A B)
6	2, 5	hbim 702	. . 3 ⊢ ((C ∈ A → D ⊆ ∪x ∈ A B) → ∀x(C ∈ A → D ⊆ ∪x ∈ A B))
7		eleq1 1149	. . . 4 ⊢ (x = C → (x ∈ A ↔ C ∈ A))
8		ssiun2s.1	. . . . 5 ⊢ (x = C → B = D)
9	8	sseq1d 1527	. . . 4 ⊢ (x = C → (B ⊆ ∪x ∈ A B ↔ D ⊆ ∪x ∈ A B))
10	7, 9	imbi12d 474	. . 3 ⊢ (x = C → ((x ∈ A → B ⊆ ∪x ∈ A B) ↔ (C ∈ A → D ⊆ ∪x ∈ A B)))
11		ssiun2 2019	. . 3 ⊢ (x ∈ A → B ⊆ ∪x ∈ A B)
12	1, 6, 10, 11	vtoclgf 1382	. 2 ⊢ (C ∈ A → (C ∈ A → D ⊆ ∪x ∈ A B))
13	12	pm2.43i 58	1 ⊢ (C ∈ A → D ⊆ ∪x ∈ A B)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∪ciun 1994

This theorem is referenced by:  oaordi 3148  omordi 3164  alephordlem2 3678  alephordi 3679

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iun 1996

metamath.org