Theorem clos1basesucg 5885

Description: A member of a closure is either in the base set or connected to another member by R. Theorem IX.5.16 of [Rosser] p. 248. (Contributed by Scott Fenton, 31-Jul-2019.)

Hypothesis

Ref	Expression
clos1basesucg.1	⊢ C = Clos1 (S, R)

Assertion

Ref	Expression
clos1basesucg	⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ C ↔ (A ∈ S ∨ ∃x ∈ C xRA)))

Distinct variable groups:   x,A   x,S   x,R   x,C

Allowed substitution hints:   V(x)   W(x)

Proof of Theorem clos1basesucg

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clos1eq1 5875	. . . . 5 ⊢ (s = S → Clos1 (s, r) = Clos1 (S, r))
2	1	eleq2d 2420	. . . 4 ⊢ (s = S → (A ∈ Clos1 (s, r) ↔ A ∈ Clos1 (S, r)))
3		eleq2 2414	. . . . 5 ⊢ (s = S → (A ∈ s ↔ A ∈ S))
4	1	rexeqdv 2815	. . . . 5 ⊢ (s = S → (∃x ∈ Clos1 (s, r)xrA ↔ ∃x ∈ Clos1 (S, r)xrA))
5	3, 4	orbi12d 690	. . . 4 ⊢ (s = S → ((A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA)))
6	2, 5	bibi12d 312	. . 3 ⊢ (s = S → ((A ∈ Clos1 (s, r) ↔ (A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA)) ↔ (A ∈ Clos1 (S, r) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA))))
7		clos1eq2 5876	. . . . 5 ⊢ (r = R → Clos1 (S, r) = Clos1 (S, R))
8	7	eleq2d 2420	. . . 4 ⊢ (r = R → (A ∈ Clos1 (S, r) ↔ A ∈ Clos1 (S, R)))
9		breq 4642	. . . . . 6 ⊢ (r = R → (xrA ↔ xRA))
10	7, 9	rexeqbidv 2821	. . . . 5 ⊢ (r = R → (∃x ∈ Clos1 (S, r)xrA ↔ ∃x ∈ Clos1 (S, R)xRA))
11	10	orbi2d 682	. . . 4 ⊢ (r = R → ((A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA)))
12	8, 11	bibi12d 312	. . 3 ⊢ (r = R → ((A ∈ Clos1 (S, r) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA)) ↔ (A ∈ Clos1 (S, R) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA))))
13		vex 2863	. . . 4 ⊢ s ∈ V
14		vex 2863	. . . 4 ⊢ r ∈ V
15		eqid 2353	. . . 4 ⊢ Clos1 (s, r) = Clos1 (s, r)
16	13, 14, 15	clos1basesuc 5883	. . 3 ⊢ (A ∈ Clos1 (s, r) ↔ (A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA))
17	6, 12, 16	vtocl2g 2919	. 2 ⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ Clos1 (S, R) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA)))
18		clos1basesucg.1	. . 3 ⊢ C = Clos1 (S, R)
19	18	eleq2i 2417	. 2 ⊢ (A ∈ C ↔ A ∈ Clos1 (S, R))
20	18	rexeqi 2813	. . 3 ⊢ (∃x ∈ C xRA ↔ ∃x ∈ Clos1 (S, R)xRA)
21	20	orbi2i 505	. 2 ⊢ ((A ∈ S ∨ ∃x ∈ C xRA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA))
22	17, 19, 21	3bitr4g 279	1 ⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ C ↔ (A ∈ S ∨ ∃x ∈ C xRA)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   class class class wbr 4640   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:  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320  frec0  6322  frecsuc  6323