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Mirrors > Home > ILE Home > Th. List > iserd | GIF version |
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.) |
Ref | Expression |
---|---|
iserd.1 | ⊢ (φ → Rel 𝑅) |
iserd.2 | ⊢ ((φ ∧ x𝑅y) → y𝑅x) |
iserd.3 | ⊢ ((φ ∧ (x𝑅y ∧ y𝑅z)) → x𝑅z) |
iserd.4 | ⊢ (φ → (x ∈ A ↔ x𝑅x)) |
Ref | Expression |
---|---|
iserd | ⊢ (φ → 𝑅 Er A) |
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) |
Colors of variables: wff set class |
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 |
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