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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 4657 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶)) | |
2 | eqeq2 2633 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶)) | |
3 | 1, 2 | bibi12d 335 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
4 | 3 | imbi2d 330 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))) |
5 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | opres 5406 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴) ↔ 〈𝐵, 𝑥〉 ∈ I )) |
7 | df-br 4654 | . . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ 〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴)) | |
8 | 5 | ideq 5274 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥) |
9 | df-br 4654 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) | |
10 | 8, 9 | bitr3i 266 | . . . 4 ⊢ (𝐵 = 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) |
11 | 6, 7, 10 | 3bitr4g 303 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) |
12 | 4, 11 | vtoclg 3266 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
13 | 12 | impcom 446 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 I cid 5023 ↾ cres 5116 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 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-id 5024 df-xp 5120 df-rel 5121 df-res 5126 |
This theorem is referenced by: foeqcnvco 6555 f1eqcocnv 6556 dfle2 11980 pospo 16973 dirref 17235 ustref 22022 trust 22033 brfvrcld2 37984 |
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