Theorem ssofss 4077

Description: Condition for subset when A is already known to be a subset. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
ssofss	|- (A C_ C -> (A C_ B <-> A.x e. C (x e. A -> x e. B)))

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

Proof of Theorem ssofss

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 |- x e. _V
2	1	elcompl 3226	. . . . . . 7 |- (x e. ∼ C <-> -. x e. C)
3		ssel 3268	. . . . . . . 8 |- (A C_ C -> (x e. A -> x e. C))
4	3	con3d 125	. . . . . . 7 |- (A C_ C -> (-. x e. C -> -. x e. A))
5	2, 4	syl5bi 208	. . . . . 6 |- (A C_ C -> (x e. ∼ C -> -. x e. A))
6	5	imp 418	. . . . 5 |- ((A C_ C /\ x e. ∼ C) -> -. x e. A)
7	6	pm2.21d 98	. . . 4 |- ((A C_ C /\ x e. ∼ C) -> (x e. A -> x e. B))
8	7	ralrimiva 2698	. . 3 |- (A C_ C -> A.x e. ∼ C(x e. A -> x e. B))
9	8	biantrud 493	. 2 |- (A C_ C -> (A.x e. C (x e. A -> x e. B) <-> (A.x e. C (x e. A -> x e. B) /\ A.x e. ∼ C(x e. A -> x e. B))))
10		ralv 2873	. . . 4 |- (A.x e. _V (x e. A -> x e. B) <-> A.x(x e. A -> x e. B))
11		uncompl 4075	. . . . 5 |- (C u. ∼ C) = _V
12	11	raleqi 2812	. . . 4 |- (A.x e. (C u. ∼ C)(x e. A -> x e. B) <-> A.x e. _V (x e. A -> x e. B))
13		dfss2 3263	. . . 4 |- (A C_ B <-> A.x(x e. A -> x e. B))
14	10, 12, 13	3bitr4ri 269	. . 3 |- (A C_ B <-> A.x e. (C u. ∼ C)(x e. A -> x e. B))
15		ralunb 3445	. . 3 |- (A.x e. (C u. ∼ C)(x e. A -> x e. B) <-> (A.x e. C (x e. A -> x e. B) /\ A.x e. ∼ C(x e. A -> x e. B)))
16	14, 15	bitri 240	. 2 |- (A C_ B <-> (A.x e. C (x e. A -> x e. B) /\ A.x e. ∼ C(x e. A -> x e. B)))
17	9, 16	syl6rbbr 255	1 |- (A C_ C -> (A C_ B <-> A.x e. C (x e. A -> x e. B)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  A.wral 2615  _Vcvv 2860   ∼ ccompl 3206   u. cun 3208   C_ wss 3258

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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  ssofeq  4078  ssrelk  4212