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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 5070 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶)) | |
2 | eqeq2 2833 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶)) | |
3 | 1, 2 | bibi12d 348 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
4 | 3 | imbi2d 343 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))) |
5 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | opres 5863 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴) ↔ 〈𝐵, 𝑥〉 ∈ I )) |
7 | df-br 5067 | . . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ 〈𝐵, 𝑥〉 ∈ ( I ↾ 𝐴)) | |
8 | 5 | ideq 5723 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥) |
9 | df-br 5067 | . . . . 5 ⊢ (𝐵 I 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) | |
10 | 8, 9 | bitr3i 279 | . . . 4 ⊢ (𝐵 = 𝑥 ↔ 〈𝐵, 𝑥〉 ∈ I ) |
11 | 6, 7, 10 | 3bitr4g 316 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) |
12 | 4, 11 | vtoclg 3567 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))) |
13 | 12 | impcom 410 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 I cid 5459 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: foeqcnvco 7056 f1eqcocnv 7057 dfle2 12541 pospo 17583 dirref 17845 ustref 22827 trust 22838 brfvrcld2 40057 |
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