Theorem gid0 8271

Description: The identity of the empty set is the empty set. (Contributed by FL, 5-Feb-2010.)

Assertion

Ref	Expression
gid0	⊢ (Id ‘∅) = ∅

Proof of Theorem gid0

Step	Hyp	Ref	Expression
1		df-gid 8250	. . 3 ⊢ Id = {<.g, y>.∣y = ∪{u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)}}
2	1	fveq1i 3836	. 2 ⊢ (Id ‘∅) = ({<.g, y>.∣y = ∪{u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)}} ‘∅)
3		0ex 2785	. . 3 ⊢ ∅ ∈ V
4		rneq 3426	. . . . . . 7 ⊢ (g = ∅ → ran g = ran ∅)
5		rabeq 1855	. . . . . . 7 ⊢ (ran g = ran ∅ → {u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = {u ∈ ran ∅∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)})
6	4, 5	syl 10	. . . . . 6 ⊢ (g = ∅ → {u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = {u ∈ ran ∅∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)})
7	4	adantr 389	. . . . . . . 8 ⊢ ((g = ∅ ⋀ u ∈ ran ∅) → ran g = ran ∅)
8	7	raleq1d 1835	. . . . . . 7 ⊢ ((g = ∅ ⋀ u ∈ ran ∅) → (∀x ∈ ran g((ugx) = x ⋀ (xgu) = x) ↔ ∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)))
9	8	rabbidv 1852	. . . . . 6 ⊢ (g = ∅ → {u ∈ ran ∅∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = {u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)})
10	6, 9	eqtrd 1550	. . . . 5 ⊢ (g = ∅ → {u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = {u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)})
11	10	unieqd 2578	. . . 4 ⊢ (g = ∅ → ∪{u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = ∪{u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)})
12		rn0 3442	. . . . . . . 8 ⊢ ran ∅ = ∅
13		rabeq 1855	. . . . . . . 8 ⊢ (ran ∅ = ∅ → {u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = {u ∈ ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)})
14	12, 13	ax-mp 7	. . . . . . 7 ⊢ {u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = {u ∈ ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)}
15		rab0 2346	. . . . . . 7 ⊢ {u ∈ ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = ∅
16	14, 15	eqtri 1538	. . . . . 6 ⊢ {u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = ∅
17	16	unieqi 2577	. . . . 5 ⊢ ∪{u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = ∪∅
18		uni0 2592	. . . . 5 ⊢ ∪∅ = ∅
19	17, 18	eqtri 1538	. . . 4 ⊢ ∪{u ∈ ran ∅∣∀x ∈ ran ∅((ugx) = x ⋀ (xgu) = x)} = ∅
20	11, 19	syl6eq 1566	. . 3 ⊢ (g = ∅ → ∪{u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)} = ∅)
21	3, 3, 20	fvopab 3901	. 2 ⊢ ({<.g, y>.∣y = ∪{u ∈ ran g∣∀x ∈ ran g((ugx) = x ⋀ (xgu) = x)}} ‘∅) = ∅
22	2, 21	eqtri 1538	1 ⊢ (Id ‘∅) = ∅

Syntax hints:   ⋀ wa 221   = wceq 992   ∈ wcel 994  ∀wral 1691  {crab 1694  ∅c0 2332  ∪cuni 2569  {copab 2740  ran crn 3252   ‘cfv 3263  (class class class)co 4021  Idcgi 8245

This theorem is referenced by:  0vfval 8472

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fv 3279  df-gid 8250

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