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Theorem resieq 6010
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 5151 . . . . 5 (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥𝐵( I ↾ 𝐴)𝐶))
2 eqeq2 2746 . . . . 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 3481 . . . . 5 𝑥 ∈ V
65opres 6009 . . . 4 (𝐵𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7 df-br 5148 . . . 4 (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
85ideq 5865 . . . . 5 (𝐵 I 𝑥𝐵 = 𝑥)
9 df-br 5148 . . . . 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 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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