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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 5493 | . 2 ⊢ Rel 𝑅 |
3 | id 22 | . . . 4 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
4 | 3 | eqcomd 2784 | . . 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 8062 | . . 3 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
7 | 5, 1 | eqerlem 8062 | . . 3 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
8 | 4, 6, 7 | 3imtr4i 284 | . 2 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧) |
9 | eqtr 2799 | . . 3 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) | |
10 | 5, 1 | eqerlem 8062 | . . . 4 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
11 | 6, 10 | anbi12i 620 | . . 3 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)) |
12 | 5, 1 | eqerlem 8062 | . . 3 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
13 | 9, 11, 12 | 3imtr4i 284 | . 2 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣) |
14 | vex 3401 | . . 3 ⊢ 𝑧 ∈ V | |
15 | eqid 2778 | . . . 4 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 | |
16 | 5, 1 | eqerlem 8062 | . . . 4 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
17 | 15, 16 | mpbir 223 | . . 3 ⊢ 𝑧𝑅𝑧 |
18 | 14, 17 | 2th 256 | . 2 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧) |
19 | 2, 8, 13, 18 | iseri 8055 | 1 ⊢ 𝑅 Er V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⦋csb 3751 class class class wbr 4888 {copab 4950 Er wer 8025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-er 8028 |
This theorem is referenced by: ider 8064 frgpuplem 18582 fneer 32944 |
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