Theorem iserd 8672

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 2738	. . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
3		iserd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
4	3	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
5		iserd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
6	5	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
7	4, 6	jca 511	. . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
8	7	alrimiv 1929	. . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9	8	alrimiv 1929	. . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
10	9	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11		dfer2 8646	. . 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 1345	. 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
15	13, 14	erref 8666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
16	15	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥))
17		vex 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
18	17, 17	breldm 5865	. . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
19	16, 18	impbid1 225	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥))
20		iserd.4	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))
21	19, 20	bitr4d 282	. . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴))
22	21	eqrdv 2735	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
23		ereq2 8654	. . 3 ⊢ (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
24	22, 23	syl 17	. 2 ⊢ (𝜑 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
25	12, 24	mpbid 232	1 ⊢ (𝜑 → 𝑅 Er 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  dom cdm 5632  Rel wrel 5637   Er wer 8642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-er 8645

This theorem is referenced by:  iseri  8673  iseriALT  8674  swoer  8677  iiner  8738  erinxp  8740  cicer  17742  eqger  19119  gaorber  19249  efgrelexlemb  19691  efgcpbllemb  19696  xmeter  24389  ercgrg  28601  erler  33358  metider  34071  prjsper  42955  cicerALT  49394