Theorem clos1induct 5881

Description: Inductive law for closure. If the base set is a subset of X, and X is closed under R, then the closure is a subset of X. Theorem IX.5.15 of [Rosser] p. 247. (Contributed by SF, 11-Feb-2015.)

Hypotheses

Ref	Expression
clos1induct.1	|- S e. _V
clos1induct.2	|- R e. _V
clos1induct.3	|- C = Clos1 (S, R)

Assertion

Ref	Expression
clos1induct	|- ((X e. V /\ S C_ X /\ A.x e. C A.z((x e. X /\ xRz) -> z e. X)) -> C C_ X)

Distinct variable groups:   x,C,z   x,R,z   x,X,z

Allowed substitution hints:   S(x,z)   V(x,z)

Proof of Theorem clos1induct

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clos1induct.3	. . . 4 |- C = Clos1 (S, R)
2		clos1induct.1	. . . . 5 |- S e. _V
3		clos1induct.2	. . . . 5 |- R e. _V
4	2, 3	clos1ex 5877	. . . 4 |- Clos1 (S, R) e. _V
5	1, 4	eqeltri 2423	. . 3 |- C e. _V
6		inexg 4101	. . 3 |- ((X e. V /\ C e. _V) -> (X i^i C) e. _V)
7	5, 6	mpan2 652	. 2 |- (X e. V -> (X i^i C) e. _V)
8	1	clos1base 5879	. . 3 |- S C_ C
9		ssin 3478	. . . 4 |- ((S C_ X /\ S C_ C) <-> S C_ (X i^i C))
10	9	biimpi 186	. . 3 |- ((S C_ X /\ S C_ C) -> S C_ (X i^i C))
11	8, 10	mpan2 652	. 2 |- (S C_ X -> S C_ (X i^i C))
12		elima2 4756	. . . . . . 7 |- (z e. (R"(X i^i C)) <-> E.x(x e. (X i^i C) /\ xRz))
13		elin 3220	. . . . . . 7 |- (z e. (X i^i C) <-> (z e. X /\ z e. C))
14	12, 13	imbi12i 316	. . . . . 6 |- ((z e. (R"(X i^i C)) -> z e. (X i^i C)) <-> (E.x(x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)))
15		df-ral 2620	. . . . . . . 8 |- (A.x e. C ((x e. X /\ xRz) -> z e. X) <-> A.x(x e. C -> ((x e. X /\ xRz) -> z e. X)))
16		impexp 433	. . . . . . . . . 10 |- (((x e. C /\ (x e. X /\ xRz)) -> z e. X) <-> (x e. C -> ((x e. X /\ xRz) -> z e. X)))
17	1	clos1conn 5880	. . . . . . . . . . . . 13 |- ((x e. C /\ xRz) -> z e. C)
18	17	biantrud 493	. . . . . . . . . . . 12 |- ((x e. C /\ xRz) -> (z e. X <-> (z e. X /\ z e. C)))
19	18	adantrl 696	. . . . . . . . . . 11 |- ((x e. C /\ (x e. X /\ xRz)) -> (z e. X <-> (z e. X /\ z e. C)))
20	19	pm5.74i 236	. . . . . . . . . 10 |- (((x e. C /\ (x e. X /\ xRz)) -> z e. X) <-> ((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
21	16, 20	bitr3i 242	. . . . . . . . 9 |- ((x e. C -> ((x e. X /\ xRz) -> z e. X)) <-> ((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
22	21	albii 1566	. . . . . . . 8 |- (A.x(x e. C -> ((x e. X /\ xRz) -> z e. X)) <-> A.x((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
23	15, 22	bitri 240	. . . . . . 7 |- (A.x e. C ((x e. X /\ xRz) -> z e. X) <-> A.x((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
24		elin 3220	. . . . . . . . . . . 12 |- (x e. (X i^i C) <-> (x e. X /\ x e. C))
25		ancom 437	. . . . . . . . . . . 12 |- ((x e. X /\ x e. C) <-> (x e. C /\ x e. X))
26	24, 25	bitri 240	. . . . . . . . . . 11 |- (x e. (X i^i C) <-> (x e. C /\ x e. X))
27	26	anbi1i 676	. . . . . . . . . 10 |- ((x e. (X i^i C) /\ xRz) <-> ((x e. C /\ x e. X) /\ xRz))
28		anass 630	. . . . . . . . . 10 |- (((x e. C /\ x e. X) /\ xRz) <-> (x e. C /\ (x e. X /\ xRz)))
29	27, 28	bitri 240	. . . . . . . . 9 |- ((x e. (X i^i C) /\ xRz) <-> (x e. C /\ (x e. X /\ xRz)))
30	29	imbi1i 315	. . . . . . . 8 |- (((x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)) <-> ((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
31	30	albii 1566	. . . . . . 7 |- (A.x((x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)) <-> A.x((x e. C /\ (x e. X /\ xRz)) -> (z e. X /\ z e. C)))
32		19.23v 1891	. . . . . . 7 |- (A.x((x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)) <-> (E.x(x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)))
33	23, 31, 32	3bitr2i 264	. . . . . 6 |- (A.x e. C ((x e. X /\ xRz) -> z e. X) <-> (E.x(x e. (X i^i C) /\ xRz) -> (z e. X /\ z e. C)))
34	14, 33	bitr4i 243	. . . . 5 |- ((z e. (R"(X i^i C)) -> z e. (X i^i C)) <-> A.x e. C ((x e. X /\ xRz) -> z e. X))
35	34	albii 1566	. . . 4 |- (A.z(z e. (R"(X i^i C)) -> z e. (X i^i C)) <-> A.zA.x e. C ((x e. X /\ xRz) -> z e. X))
36		dfss2 3263	. . . 4 |- ((R"(X i^i C)) C_ (X i^i C) <-> A.z(z e. (R"(X i^i C)) -> z e. (X i^i C)))
37		ralcom4 2878	. . . 4 |- (A.x e. C A.z((x e. X /\ xRz) -> z e. X) <-> A.zA.x e. C ((x e. X /\ xRz) -> z e. X))
38	35, 36, 37	3bitr4i 268	. . 3 |- ((R"(X i^i C)) C_ (X i^i C) <-> A.x e. C A.z((x e. X /\ xRz) -> z e. X))
39	38	biimpri 197	. 2 |- (A.x e. C A.z((x e. X /\ xRz) -> z e. X) -> (R"(X i^i C)) C_ (X i^i C))
40		df-clos1 5874	. . . . 5 |- Clos1 (S, R) = |^|{a | (S C_ a /\ (R"a) C_ a)}
41	1, 40	eqtri 2373	. . . 4 |- C = |^|{a | (S C_ a /\ (R"a) C_ a)}
42		sseq2 3294	. . . . . . . . 9 |- (a = (X i^i C) -> (S C_ a <-> S C_ (X i^i C)))
43		imaeq2 4939	. . . . . . . . . 10 |- (a = (X i^i C) -> (R"a) = (R"(X i^i C)))
44		id 19	. . . . . . . . . 10 |- (a = (X i^i C) -> a = (X i^i C))
45	43, 44	sseq12d 3301	. . . . . . . . 9 |- (a = (X i^i C) -> ((R"a) C_ a <-> (R"(X i^i C)) C_ (X i^i C)))
46	42, 45	anbi12d 691	. . . . . . . 8 |- (a = (X i^i C) -> ((S C_ a /\ (R"a) C_ a) <-> (S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C))))
47	46	elabg 2987	. . . . . . 7 |- ((X i^i C) e. _V -> ((X i^i C) e. {a | (S C_ a /\ (R"a) C_ a)} <-> (S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C))))
48	47	biimprd 214	. . . . . 6 |- ((X i^i C) e. _V -> ((S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C)) -> (X i^i C) e. {a | (S C_ a /\ (R"a) C_ a)}))
49	48	3impib 1149	. . . . 5 |- (((X i^i C) e. _V /\ S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C)) -> (X i^i C) e. {a | (S C_ a /\ (R"a) C_ a)})
50		intss1 3942	. . . . 5 |- ((X i^i C) e. {a | (S C_ a /\ (R"a) C_ a)} -> |^|{a | (S C_ a /\ (R"a) C_ a)} C_ (X i^i C))
51	49, 50	syl 15	. . . 4 |- (((X i^i C) e. _V /\ S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C)) -> |^|{a | (S C_ a /\ (R"a) C_ a)} C_ (X i^i C))
52	41, 51	syl5eqss 3316	. . 3 |- (((X i^i C) e. _V /\ S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C)) -> C C_ (X i^i C))
53		inss1 3476	. . 3 |- (X i^i C) C_ X
54	52, 53	syl6ss 3285	. 2 |- (((X i^i C) e. _V /\ S C_ (X i^i C) /\ (R"(X i^i C)) C_ (X i^i C)) -> C C_ X)
55	7, 11, 39, 54	syl3an 1224	1 |- ((X e. V /\ S C_ X /\ A.x e. C A.z((x e. X /\ xRz) -> z e. X)) -> C C_ X)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  _Vcvv 2860   i^i cin 3209   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-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-clos1 5874

This theorem is referenced by:  clos1is  5882  clos1nrel  5887  clos10  5888  spacind  6288  frecxp  6315