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Mirrors > Home > MPE Home > Th. List > eqerOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of eqer 7777 as of 1-May-2021. 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 modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eqer.1 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
eqer.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} |
Ref | Expression |
---|---|
eqerOLD | ⊢ 𝑅 Er V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqer.2 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} | |
2 | 1 | relopabi 5245 | . . . 4 ⊢ Rel 𝑅 |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Rel 𝑅) |
4 | id 22 | . . . . . 6 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
5 | 4 | eqcomd 2628 | . . . . 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 7776 | . . . . 5 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
8 | 6, 1 | eqerlem 7776 | . . . . 5 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
9 | 5, 7, 8 | 3imtr4i 281 | . . . 4 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧) |
10 | 9 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧) |
11 | eqtr 2641 | . . . . 5 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) | |
12 | 6, 1 | eqerlem 7776 | . . . . . 6 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
13 | 7, 12 | anbi12i 733 | . . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)) |
14 | 6, 1 | eqerlem 7776 | . . . . 5 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) |
15 | 11, 13, 14 | 3imtr4i 281 | . . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣) |
16 | 15 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣)) → 𝑧𝑅𝑣) |
17 | vex 3203 | . . . . 5 ⊢ 𝑧 ∈ V | |
18 | eqid 2622 | . . . . . 6 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 | |
19 | 6, 1 | eqerlem 7776 | . . . . . 6 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
20 | 18, 19 | mpbir 221 | . . . . 5 ⊢ 𝑧𝑅𝑧 |
21 | 17, 20 | 2th 254 | . . . 4 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧) |
22 | 21 | a1i 11 | . . 3 ⊢ (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧)) |
23 | 3, 10, 16, 22 | iserd 7768 | . 2 ⊢ (⊤ → 𝑅 Er V) |
24 | 23 | trud 1493 | 1 ⊢ 𝑅 Er V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 Vcvv 3200 ⦋csb 3533 class class class wbr 4653 {copab 4712 Rel wrel 5119 Er wer 7739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-er 7742 |
This theorem is referenced by: (None) |
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