Theorem iserd 8772

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 2764	. . 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 1962	. . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9	8	alrimiv 1962	. . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
10	9	alrimiv 1962	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11		dfer2 8746	. . 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 8766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
16	15	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥))
17		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
18	17, 17	breldm 5897	. . . . . 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 2761	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
23		ereq2 8754	. . 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 2147   class class class wbr 5106  dom cdm 5655  Rel wrel 5660   Er wer 8742

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-er 8745

This theorem is used by:  iseri  8773  iseriALT  8774  swoer  8777  iiner  8838  erinxp  8840  cicer  18010  eqger  19419  gaorber  19552  efgrelexlemb  19994  efgcpbllemb  19999  xmeter  24782  ercgrg  29010  cgraer  29407  erler  33896  metider  34596  prjsper  43829  cicerALT  50365