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Theorem resieq 6007
Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
resieq ((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))

Proof of Theorem resieq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq2 5146 . . . . 5 (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥𝐵( I ↾ 𝐴)𝐶))
2 eqeq2 2748 . . . . 5 (𝑥 = 𝐶 → (𝐵 = 𝑥𝐵 = 𝐶))
31, 2bibi12d 345 . . . 4 (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
43imbi2d 340 . . 3 (𝑥 = 𝐶 → ((𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥)) ↔ (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))))
5 vex 3483 . . . . 5 𝑥 ∈ V
65opres 6006 . . . 4 (𝐵𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7 df-br 5143 . . . 4 (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
85ideq 5862 . . . . 5 (𝐵 I 𝑥𝐵 = 𝑥)
9 df-br 5143 . . . . 5 (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
108, 9bitr3i 277 . . . 4 (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
116, 7, 103bitr4g 314 . . 3 (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥))
124, 11vtoclg 3553 . 2 (𝐶𝐴 → (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
1312impcom 407 1 ((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cop 4631   class class class wbr 5142   I cid 5576  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-res 5696
This theorem is referenced by:  foeqcnvco  7321  f1eqcocnv  7322  dfle2  13190  pospo  18391  dirref  18647  ustref  24228  trust  24239  brfvrcld2  43710  resipos  48879
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