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Theorem resieq 5442
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 4689 . . . . 5 (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥𝐵( I ↾ 𝐴)𝐶))
2 eqeq2 2662 . . . . 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 3234 . . . . 5 𝑥 ∈ V
65opres 5441 . . . 4 (𝐵𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7 df-br 4686 . . . 4 (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
85ideq 5307 . . . . 5 (𝐵 I 𝑥𝐵 = 𝑥)
9 df-br 4686 . . . . 5 (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
108, 9bitr3i 266 . . . 4 (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
116, 7, 103bitr4g 303 . . 3 (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥))
124, 11vtoclg 3297 . 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 196  wa 383   = wceq 1523  wcel 2030  cop 4216   class class class wbr 4685   I cid 5052  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  foeqcnvco  6595  f1eqcocnv  6596  dfle2  12018  pospo  17020  dirref  17282  ustref  22069  trust  22080  brfvrcld2  38301
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