Theorem clos1nrel 5886

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 3564	. . . . 5 ⊢ ((R “ S) = ∅ ↔ ∀y ¬ y ∈ (R “ S))
2		rspe 2675	. . . . . . . . 9 ⊢ ((x ∈ S ∧ xRy) → ∃x ∈ S xRy)
3		elima 4754	. . . . . . . . 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 2698	. . 3 ⊢ ((R “ S) = ∅ → ∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S))
10		clos1nrel.1	. . . 4 ⊢ S ∈ V
11		ssid 3290	. . . 4 ⊢ S ⊆ S
12		clos1nrel.2	. . . . 5 ⊢ R ∈ V
13		clos1nrel.3	. . . . 5 ⊢ C = Clos1 (S, R)
14	10, 12, 13	clos1induct 5880	. . . 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 5878	. . 3 ⊢ S ⊆ C
18	17	a1i 10	. 2 ⊢ ((R “ S) = ∅ → S ⊆ C)
19	16, 18	eqssd 3289	1 ⊢ ((R “ S) = ∅ → C = S)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∅c0 3550   class class class wbr 4639   “ cima 4722   Clos1 cclos1 5872

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873

This theorem is referenced by:  nchoicelem3  6291