Theorem iserd 6068

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 2038	. . 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 1751	. . . . 5 ⊢ (φ → ∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
9	8	alrimiv 1751	. . . 4 ⊢ (φ → ∀y∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
10	9	alrimiv 1751	. . 3 ⊢ (φ → ∀x∀y∀z((x𝑅y → y𝑅x) ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
11		dfer2 6043	. . 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 1087	. 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 6062	. . . . . . 7 ⊢ ((φ ∧ x ∈ dom 𝑅) → x𝑅x)
16	15	ex 108	. . . . . 6 ⊢ (φ → (x ∈ dom 𝑅 → x𝑅x))
17		vex 2554	. . . . . . 7 ⊢ x ∈ V
18	17, 17	breldm 4482	. . . . . 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 2035	. . 3 ⊢ (φ → dom 𝑅 = A)
23		ereq2 6050	. . 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 1240   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  dom cdm 4288  Rel wrel 4293   Er wer 6039

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042

This theorem is referenced by:  swoer  6070  eqer  6074  0er  6076  iinerm  6114  erinxp  6116  ecopover  6140  ecopoverg  6143  ener  6195  enq0er  6418