Theorem uniiunlem 3353

Description: A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Assertion

Ref	Expression
uniiunlem	⊢ (∀x ∈ A B ∈ D → (∀x ∈ A B ∈ C ↔ {y ∣ ∃x ∈ A y = B} ⊆ C))

Distinct variable groups:   x,y   y,A   y,B   x,C

Allowed substitution hints:   A(x)   B(x)   C(y)   D(x,y)

Proof of Theorem uniiunlem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . 6 ⊢ (y = z → (y = B ↔ z = B))
2	1	rexbidv 2635	. . . . 5 ⊢ (y = z → (∃x ∈ A y = B ↔ ∃x ∈ A z = B))
3	2	cbvabv 2472	. . . 4 ⊢ {y ∣ ∃x ∈ A y = B} = {z ∣ ∃x ∈ A z = B}
4	3	sseq1i 3295	. . 3 ⊢ ({y ∣ ∃x ∈ A y = B} ⊆ C ↔ {z ∣ ∃x ∈ A z = B} ⊆ C)
5		r19.23v 2730	. . . . 5 ⊢ (∀x ∈ A (z = B → z ∈ C) ↔ (∃x ∈ A z = B → z ∈ C))
6	5	albii 1566	. . . 4 ⊢ (∀z∀x ∈ A (z = B → z ∈ C) ↔ ∀z(∃x ∈ A z = B → z ∈ C))
7		ralcom4 2877	. . . 4 ⊢ (∀x ∈ A ∀z(z = B → z ∈ C) ↔ ∀z∀x ∈ A (z = B → z ∈ C))
8		abss 3335	. . . 4 ⊢ ({z ∣ ∃x ∈ A z = B} ⊆ C ↔ ∀z(∃x ∈ A z = B → z ∈ C))
9	6, 7, 8	3bitr4i 268	. . 3 ⊢ (∀x ∈ A ∀z(z = B → z ∈ C) ↔ {z ∣ ∃x ∈ A z = B} ⊆ C)
10	4, 9	bitr4i 243	. 2 ⊢ ({y ∣ ∃x ∈ A y = B} ⊆ C ↔ ∀x ∈ A ∀z(z = B → z ∈ C))
11		nfv 1619	. . . . 5 ⊢ Ⅎz B ∈ C
12		eleq1 2413	. . . . 5 ⊢ (z = B → (z ∈ C ↔ B ∈ C))
13	11, 12	ceqsalg 2883	. . . 4 ⊢ (B ∈ D → (∀z(z = B → z ∈ C) ↔ B ∈ C))
14	13	ralimi 2689	. . 3 ⊢ (∀x ∈ A B ∈ D → ∀x ∈ A (∀z(z = B → z ∈ C) ↔ B ∈ C))
15		ralbi 2750	. . 3 ⊢ (∀x ∈ A (∀z(z = B → z ∈ C) ↔ B ∈ C) → (∀x ∈ A ∀z(z = B → z ∈ C) ↔ ∀x ∈ A B ∈ C))
16	14, 15	syl 15	. 2 ⊢ (∀x ∈ A B ∈ D → (∀x ∈ A ∀z(z = B → z ∈ C) ↔ ∀x ∈ A B ∈ C))
17	10, 16	syl5rbb 249	1 ⊢ (∀x ∈ A B ∈ D → (∀x ∈ A B ∈ C ↔ {y ∣ ∃x ∈ A y = B} ⊆ C))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)