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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-idreseqb | Structured version Visualization version GIF version |
Description: Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.) |
Ref | Expression |
---|---|
bj-idreseqb | ⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5847 | . . 3 ⊢ Rel ( I ↾ 𝐶) | |
2 | 1 | brrelex12i 5571 | . 2 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | |
4 | 3 | elexd 3461 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
5 | eleq1 2877 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
6 | 5 | biimpac 482 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶) |
7 | 6 | elexd 3461 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
8 | 4, 7 | jca 515 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | brres 5825 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵))) | |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵))) |
11 | eqeq12 2812 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) | |
12 | df-id 5425 | . . . . 5 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
13 | 11, 12 | brabga 5386 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
14 | 13 | anbi2d 631 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))) |
15 | 10, 14 | bitrd 282 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))) |
16 | 2, 8, 15 | pm5.21nii 383 | 1 ⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 I cid 5424 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: bj-elid7 34586 |
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