Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqer | Structured version Visualization version 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.) (Proof shortened by AV, 1-May-2021.) |
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 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} | |
2 | 1 | relopabi 5167 | . 2 ⊢ Rel 𝑅 |
3 | id 22 | . . . 4 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
4 | 3 | eqcomd 2616 | . . 3 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
5 | eqer.1 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | 5, 1 | eqerlem 7663 | . . 3 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
7 | 5, 1 | eqerlem 7663 | . . 3 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
8 | 4, 6, 7 | 3imtr4i 280 | . 2 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧) |
9 | eqtr 2629 | . . 3 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) | |
10 | 5, 1 | eqerlem 7663 | . . . 4 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
11 | 6, 10 | anbi12i 729 | . . 3 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)) |
12 | 5, 1 | eqerlem 7663 | . . 3 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
13 | 9, 11, 12 | 3imtr4i 280 | . 2 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣) |
14 | vex 3176 | . . 3 ⊢ 𝑧 ∈ V | |
15 | eqid 2610 | . . . 4 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 | |
16 | 5, 1 | eqerlem 7663 | . . . 4 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
17 | 15, 16 | mpbir 220 | . . 3 ⊢ 𝑧𝑅𝑧 |
18 | 14, 17 | 2th 253 | . 2 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧) |
19 | 2, 8, 13, 18 | iseri 7656 | 1 ⊢ 𝑅 Er V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 class class class wbr 4583 {copab 4642 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-er 7629 |
This theorem is referenced by: ider 7666 frgpuplem 18008 fneer 31518 |
Copyright terms: Public domain | W3C validator |