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 ∈ V
clos1induct.2	⊢ R ∈ V
clos1induct.3	⊢ C = Clos1 (S, R)

Assertion

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

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ∩cint 3927   class class class wbr 4640   “ cima 4723   Clos1 cclos1 5873

This proof depends on 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 proof 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 used by:  clos1is  5882  clos1nrel  5887  clos10  5888  spacind  6288  frecxp  6315