Theorem clos1conn 5880

Description: If a class is connected to an element of a closure via R, then it is a member of the closure. Theorem IX.5.14 of [Rosser] p. 246. (Contributed by SF, 13-Feb-2015.)

Hypothesis

Ref	Expression
clos1base.1	|- C = Clos1 (S, R)

Assertion

Ref	Expression
clos1conn	|- ((A e. C /\ ARB) -> B e. C)

Proof of Theorem clos1conn

Dummy variables a x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 |- (ARB -> (A e. _V /\ B e. _V))
2	1	adantl 452	. 2 |- ((A e. C /\ ARB) -> (A e. _V /\ B e. _V))
3		eleq1 2413	. . . . 5 |- (x = A -> (x e. C <-> A e. C))
4		breq1 4643	. . . . 5 |- (x = A -> (xRy <-> ARy))
5	3, 4	anbi12d 691	. . . 4 |- (x = A -> ((x e. C /\ xRy) <-> (A e. C /\ ARy)))
6	5	imbi1d 308	. . 3 |- (x = A -> (((x e. C /\ xRy) -> y e. C) <-> ((A e. C /\ ARy) -> y e. C)))
7		breq2 4644	. . . . 5 |- (y = B -> (ARy <-> ARB))
8	7	anbi2d 684	. . . 4 |- (y = B -> ((A e. C /\ ARy) <-> (A e. C /\ ARB)))
9		eleq1 2413	. . . 4 |- (y = B -> (y e. C <-> B e. C))
10	8, 9	imbi12d 311	. . 3 |- (y = B -> (((A e. C /\ ARy) -> y e. C) <-> ((A e. C /\ ARB) -> B e. C)))
11		breq1 4643	. . . . . . . . . . . . . 14 |- (z = x -> (zRy <-> xRy))
12	11	rspcev 2956	. . . . . . . . . . . . 13 |- ((x e. a /\ xRy) -> E.z e. a zRy)
13		elima 4755	. . . . . . . . . . . . 13 |- (y e. (R"a) <-> E.z e. a zRy)
14	12, 13	sylibr 203	. . . . . . . . . . . 12 |- ((x e. a /\ xRy) -> y e. (R"a))
15	14	ancoms 439	. . . . . . . . . . 11 |- ((xRy /\ x e. a) -> y e. (R"a))
16		ssel 3268	. . . . . . . . . . 11 |- ((R"a) C_ a -> (y e. (R"a) -> y e. a))
17	15, 16	syl5 28	. . . . . . . . . 10 |- ((R"a) C_ a -> ((xRy /\ x e. a) -> y e. a))
18	17	exp3a 425	. . . . . . . . 9 |- ((R"a) C_ a -> (xRy -> (x e. a -> y e. a)))
19	18	com12 27	. . . . . . . 8 |- (xRy -> ((R"a) C_ a -> (x e. a -> y e. a)))
20	19	adantld 453	. . . . . . 7 |- (xRy -> ((S C_ a /\ (R"a) C_ a) -> (x e. a -> y e. a)))
21	20	a2d 23	. . . . . 6 |- (xRy -> (((S C_ a /\ (R"a) C_ a) -> x e. a) -> ((S C_ a /\ (R"a) C_ a) -> y e. a)))
22	21	alimdv 1621	. . . . 5 |- (xRy -> (A.a((S C_ a /\ (R"a) C_ a) -> x e. a) -> A.a((S C_ a /\ (R"a) C_ a) -> y e. a)))
23		clos1base.1	. . . . . . . 8 |- C = Clos1 (S, R)
24		df-clos1 5874	. . . . . . . 8 |- Clos1 (S, R) = |^|{a | (S C_ a /\ (R"a) C_ a)}
25	23, 24	eqtri 2373	. . . . . . 7 |- C = |^|{a | (S C_ a /\ (R"a) C_ a)}
26	25	eleq2i 2417	. . . . . 6 |- (x e. C <-> x e. |^|{a | (S C_ a /\ (R"a) C_ a)})
27		vex 2863	. . . . . . 7 |- x e. _V
28	27	elintab 3938	. . . . . 6 |- (x e. |^|{a | (S C_ a /\ (R"a) C_ a)} <-> A.a((S C_ a /\ (R"a) C_ a) -> x e. a))
29	26, 28	bitri 240	. . . . 5 |- (x e. C <-> A.a((S C_ a /\ (R"a) C_ a) -> x e. a))
30	25	eleq2i 2417	. . . . . 6 |- (y e. C <-> y e. |^|{a | (S C_ a /\ (R"a) C_ a)})
31		vex 2863	. . . . . . 7 |- y e. _V
32	31	elintab 3938	. . . . . 6 |- (y e. |^|{a | (S C_ a /\ (R"a) C_ a)} <-> A.a((S C_ a /\ (R"a) C_ a) -> y e. a))
33	30, 32	bitri 240	. . . . 5 |- (y e. C <-> A.a((S C_ a /\ (R"a) C_ a) -> y e. a))
34	22, 29, 33	3imtr4g 261	. . . 4 |- (xRy -> (x e. C -> y e. C))
35	34	impcom 419	. . 3 |- ((x e. C /\ xRy) -> y e. C)
36	6, 10, 35	vtocl2g 2919	. 2 |- ((A e. _V /\ B e. _V) -> ((A e. C /\ ARB) -> B e. C))
37	2, 36	mpcom 32	1 |- ((A e. C /\ ARB) -> B e. C)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   C_ wss 3258  |^|cint 3927   class class class wbr 4640  "cima 4723   Clos1 cclos1 5873

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-ima 4728  df-clos1 5874

This theorem is referenced by:  clos1induct  5881  clos1basesuc  5883  spaccl  6287  dmfrec  6317