Theorem iserd 6043

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 2023	. . 3 ⊢ (φ → dom 𝑅 = dom 𝑅)
3		iserd.2	. . . . . . . 8 ⊢ ((φ ∧ x𝑅y) → y𝑅x)
4	3	ex 108	. . . . . . 7 ⊢ (φ → (x𝑅y → y𝑅x))
5		iserd.3	. . . . . . . 8 ⊢ ((φ ∧ (x𝑅y ∧ y𝑅z)) → x𝑅z)
6	5	ex 108	. . . . . . 7 ⊢ (φ → ((x𝑅y ∧ y𝑅z) → x𝑅z))
7	4, 6	jca 290	. . . . . 6 ⊢ (φ → ((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
8	7	alrimiv 1736	. . . . 5 ⊢ (φ → ∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
9	8	alrimiv 1736	. . . 4 ⊢ (φ → ∀y∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
10	9	alrimiv 1736	. . 3 ⊢ (φ → ∀x∀y∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
11		dfer2 6018	. . 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 1075	. 2 ⊢ (φ → 𝑅 Er dom 𝑅)
13	12	adantr 261	. . . . . . . 8 ⊢ ((φ ∧ x ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14		simpr 103	. . . . . . . 8 ⊢ ((φ ∧ x ∈ dom 𝑅) → x ∈ dom 𝑅)
15	13, 14	erref 6037	. . . . . . 7 ⊢ ((φ ∧ x ∈ dom 𝑅) → x𝑅x)
16	15	ex 108	. . . . . 6 ⊢ (φ → (x ∈ dom 𝑅 → x𝑅x))
17		vex 2538	. . . . . . 7 ⊢ x ∈ V
18	17, 17	breldm 4466	. . . . . 6 ⊢ (x𝑅x → x ∈ dom 𝑅)
19	16, 18	impbid1 130	. . . . 5 ⊢ (φ → (x ∈ dom 𝑅 ↔ x𝑅x))
20		iserd.4	. . . . 5 ⊢ (φ → (x ∈ A ↔ x𝑅x))
21	19, 20	bitr4d 180	. . . 4 ⊢ (φ → (x ∈ dom 𝑅 ↔ x ∈ A))
22	21	eqrdv 2020	. . 3 ⊢ (φ → dom 𝑅 = A)
23		ereq2 6025	. . 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 135	1 ⊢ (φ → 𝑅 Er A)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374   class class class wbr 3738  dom cdm 4272  Rel wrel 4277   Er wer 6014

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-er 6017

This theorem is referenced by:  swoer  6045  eqer  6049  0er  6051  iinerm  6089  erinxp  6091  ecopover  6115  ecopoverg  6118  enq0er  6290