Theorem clos1nrel 5887

Description: The value of a closure when the base set is not related to anything in R. (Contributed by SF, 13-Mar-2015.)

Hypotheses

Ref	Expression
clos1nrel.1	⊢ S ∈ V
clos1nrel.2	⊢ R ∈ V
clos1nrel.3	⊢ C = Clos1 (S, R)

Assertion

Ref	Expression
clos1nrel	⊢ ((R “ S) = ∅ → C = S)

Proof of Theorem clos1nrel

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

Step	Hyp	Ref	Expression
1		eq0 3565	. . . . 5 ⊢ ((R “ S) = ∅ ↔ ∀y ¬ y ∈ (R “ S))
2		rspe 2676	. . . . . . . . 9 ⊢ ((x ∈ S ∧ xRy) → ∃x ∈ S xRy)
3		elima 4755	. . . . . . . . 9 ⊢ (y ∈ (R “ S) ↔ ∃x ∈ S xRy)
4	2, 3	sylibr 203	. . . . . . . 8 ⊢ ((x ∈ S ∧ xRy) → y ∈ (R “ S))
5	4	con3i 127	. . . . . . 7 ⊢ (¬ y ∈ (R “ S) → ¬ (x ∈ S ∧ xRy))
6	5	pm2.21d 98	. . . . . 6 ⊢ (¬ y ∈ (R “ S) → ((x ∈ S ∧ xRy) → y ∈ S))
7	6	alimi 1559	. . . . 5 ⊢ (∀y ¬ y ∈ (R “ S) → ∀y((x ∈ S ∧ xRy) → y ∈ S))
8	1, 7	sylbi 187	. . . 4 ⊢ ((R “ S) = ∅ → ∀y((x ∈ S ∧ xRy) → y ∈ S))
9	8	ralrimivw 2699	. . 3 ⊢ ((R “ S) = ∅ → ∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S))
10		clos1nrel.1	. . . 4 ⊢ S ∈ V
11		ssid 3291	. . . 4 ⊢ S ⊆ S
12		clos1nrel.2	. . . . 5 ⊢ R ∈ V
13		clos1nrel.3	. . . . 5 ⊢ C = Clos1 (S, R)
14	10, 12, 13	clos1induct 5881	. . . 4 ⊢ ((S ∈ V ∧ S ⊆ S ∧ ∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S)) → C ⊆ S)
15	10, 11, 14	mp3an12 1267	. . 3 ⊢ (∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S) → C ⊆ S)
16	9, 15	syl 15	. 2 ⊢ ((R “ S) = ∅ → C ⊆ S)
17	13	clos1base 5879	. . 3 ⊢ S ⊆ C
18	17	a1i 10	. 2 ⊢ ((R “ S) = ∅ → S ⊆ C)
19	16, 18	eqssd 3290	1 ⊢ ((R “ S) = ∅ → C = S)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  ∅c0 3551   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:  nchoicelem3  6292