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Mirrors > Home > ILE Home > Th. List > eqer | GIF version |
Description: Equivalence relation involving equality of dependent classes 𝐴(𝑥) and 𝐵(𝑦). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
eqer.1 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
eqer.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} |
Ref | Expression |
---|---|
eqer | ⊢ 𝑅 Er V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqer.2 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} | |
2 | 1 | relopabi 4673 | . . . 4 ⊢ Rel 𝑅 |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Rel 𝑅) |
4 | id 19 | . . . . . 6 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
5 | 4 | eqcomd 2146 | . . . . 5 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
6 | eqer.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
7 | 6, 1 | eqerlem 6468 | . . . . 5 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
8 | 6, 1 | eqerlem 6468 | . . . . 5 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
9 | 5, 7, 8 | 3imtr4i 200 | . . . 4 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧) |
10 | 9 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧) |
11 | eqtr 2158 | . . . . 5 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) | |
12 | 6, 1 | eqerlem 6468 | . . . . . 6 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
13 | 7, 12 | anbi12i 456 | . . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)) |
14 | 6, 1 | eqerlem 6468 | . . . . 5 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
15 | 11, 13, 14 | 3imtr4i 200 | . . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣) |
16 | 15 | adantl 275 | . . 3 ⊢ ((⊤ ∧ (𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣)) → 𝑧𝑅𝑣) |
17 | vex 2692 | . . . . 5 ⊢ 𝑧 ∈ V | |
18 | eqid 2140 | . . . . . 6 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 | |
19 | 6, 1 | eqerlem 6468 | . . . . . 6 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
20 | 18, 19 | mpbir 145 | . . . . 5 ⊢ 𝑧𝑅𝑧 |
21 | 17, 20 | 2th 173 | . . . 4 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧) |
22 | 21 | a1i 9 | . . 3 ⊢ (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧)) |
23 | 3, 10, 16, 22 | iserd 6463 | . 2 ⊢ (⊤ → 𝑅 Er V) |
24 | 23 | mptru 1341 | 1 ⊢ 𝑅 Er V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ⊤wtru 1333 ∈ wcel 1481 Vcvv 2689 ⦋csb 3007 class class class wbr 3937 {copab 3996 Rel wrel 4552 Er wer 6434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-er 6437 |
This theorem is referenced by: ider 6470 |
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