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Mirrors > Home > MPE Home > Th. List > resieq | Structured version Visualization version GIF version |
Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
resieq | ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5061 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶)) | |
2 | eqeq2 2830 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶)) | |
3 | 1, 2 | bibi12d 347 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
4 | 3 | imbi2d 342 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))) |
5 | vex 3495 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | opres 5856 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴) ↔ 〈𝐵, 𝑥〉 ∈ I )) |
7 | df-br 5058 | . . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ 〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴)) | |
8 | 5 | ideq 5716 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥) |
9 | df-br 5058 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) | |
10 | 8, 9 | bitr3i 278 | . . . 4 ⊢ (𝐵 = 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) |
11 | 6, 7, 10 | 3bitr4g 315 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) |
12 | 4, 11 | vtoclg 3565 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
13 | 12 | impcom 408 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 I cid 5452 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: foeqcnvco 7047 f1eqcocnv 7048 dfle2 12528 pospo 17571 dirref 17833 ustref 22754 trust 22765 brfvrcld2 39915 |
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