Theorem iserd 6827

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 2239	. . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
3		iserd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
4	3	ex 115	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
5		iserd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
6	5	ex 115	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
7	4, 6	jca 306	. . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
8	7	alrimiv 1927	. . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9	8	alrimiv 1927	. . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
10	9	alrimiv 1927	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11		dfer2 6802	. . 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 1212	. 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅)
13	12	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
15	13, 14	erref 6821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
16	15	ex 115	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥))
17		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
18	17, 17	breldm 4983	. . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
19	16, 18	impbid1 142	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥))
20		iserd.4	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))
21	19, 20	bitr4d 191	. . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴))
22	21	eqrdv 2236	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
23		ereq2 6809	. . 3 ⊢ (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
24	22, 23	syl 14	. 2 ⊢ (𝜑 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴))
25	12, 24	mpbid 147	1 ⊢ (𝜑 → 𝑅 Er 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209   class class class wbr 4128  dom cdm 4772  Rel wrel 4777   Er wer 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-er 6801

This theorem is referenced by:  swoer  6829  eqer  6833  0er  6835  iinerm  6875  erinxp  6877  ecopover  6901  ecopoverg  6904  ener  7060  enq0er  7796  eqger  14010  xmeter  15520