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Theorem resieq 5327
 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 4587 . . . . 5 (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥𝐵( I ↾ 𝐴)𝐶))
2 eqeq2 2621 . . . . 5 (𝑥 = 𝐶 → (𝐵 = 𝑥𝐵 = 𝐶))
31, 2bibi12d 334 . . . 4 (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
43imbi2d 329 . . 3 (𝑥 = 𝐶 → ((𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥)) ↔ (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))))
5 vex 3176 . . . . 5 𝑥 ∈ V
65opres 5326 . . . 4 (𝐵𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7 df-br 4584 . . . 4 (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
85ideq 5196 . . . . 5 (𝐵 I 𝑥𝐵 = 𝑥)
9 df-br 4584 . . . . 5 (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
108, 9bitr3i 265 . . . 4 (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
116, 7, 103bitr4g 302 . . 3 (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥))
124, 11vtoclg 3239 . 2 (𝐶𝐴 → (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
1312impcom 445 1 ((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583   I cid 4948   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050 This theorem is referenced by:  foeqcnvco  6455  f1eqcocnv  6456  dfle2  11856  pospo  16796  dirref  17058  ustref  21832  trust  21843  brfvrcld2  37003
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