Theorem iserd 8726

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
iserd.1	⊢ (𝜑 → Rel 𝑅)
iserd.2	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
iserd.3	⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
iserd.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))

Assertion

Ref	Expression
iserd	⊢ (𝜑 → 𝑅 Er 𝐴)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑦, 𝑧)

Proof of Theorem iserd

Step	Hyp	Ref	Expression
1		iserd.1	. . 3 ⊢ (𝜑 → Rel 𝑅)
2		eqidd 2763	. . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
3		iserd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
4	3	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
5		iserd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
6	5	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
7	4, 6	jca 521	. . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
8	7	alrimiv 1960	. . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9	8	alrimiv 1960	. . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
10	9	alrimiv 1960	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11		dfer2 8700	. . 3 ⊢ (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
12	1, 2, 10, 11	syl3anbrc 1362	. 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅)
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
15	13, 14	erref 8720	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
16	15	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥))
17		vex 3457	. . . . . . 7 ⊢ 𝑥 ∈ V
18	17, 17	breldm 5896	. . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
19	16, 18	impbid1 228	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥))
20		iserd.4	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))
21	19, 20	bitr4d 285	. . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴))
22	21	eqrdv 2760	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
23		ereq2 8708	. . 3 ⊢ (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
24	22, 23	syl 18	. 2 ⊢ (𝜑 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
25	12, 24	mpbid 235	1 ⊢ (𝜑 → 𝑅 Er 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145   class class class wbr 5107  dom cdm 5659  Rel wrel 5664   Er wer 8696

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-er 8699

This theorem is used by:  iseri  8727  iseriALT  8728  swoer  8731  iiner  8792  erinxp  8794  cicer  17899  eqger  19307  gaorber  19439  efgrelexlemb  19881  efgcpbllemb  19886  xmeter  24663  ercgrg  28860  cgraer  29257  erler  33707  metider  34406  prjsper  43456  cicerALT  49974