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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 5151 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶)) | |
2 | eqeq2 2746 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶)) | |
3 | 1, 2 | bibi12d 345 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))) |
5 | vex 3481 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | opres 6009 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴) ↔ 〈𝐵, 𝑥〉 ∈ I )) |
7 | df-br 5148 | . . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ 〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴)) | |
8 | 5 | ideq 5865 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥) |
9 | df-br 5148 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) | |
10 | 8, 9 | bitr3i 277 | . . . 4 ⊢ (𝐵 = 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) |
11 | 6, 7, 10 | 3bitr4g 314 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) |
12 | 4, 11 | vtoclg 3553 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
13 | 12 | impcom 407 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 〈cop 4636 class class class wbr 5147 I cid 5581 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-res 5700 |
This theorem is referenced by: foeqcnvco 7319 f1eqcocnv 7320 dfle2 13185 pospo 18402 dirref 18658 ustref 24242 trust 24253 brfvrcld2 43681 |
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