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Theorem resieq 4901
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 3993 . . . . 5 (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥𝐵( I ↾ 𝐴)𝐶))
2 eqeq2 2180 . . . . 5 (𝑥 = 𝐶 → (𝐵 = 𝑥𝐵 = 𝐶))
31, 2bibi12d 234 . . . 4 (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
43imbi2d 229 . . 3 (𝑥 = 𝐶 → ((𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥)) ↔ (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))))
5 vex 2733 . . . . 5 𝑥 ∈ V
65opres 4900 . . . 4 (𝐵𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7 df-br 3990 . . . 4 (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
85ideq 4763 . . . . 5 (𝐵 I 𝑥𝐵 = 𝑥)
9 df-br 3990 . . . . 5 (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
108, 9bitr3i 185 . . . 4 (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
116, 7, 103bitr4g 222 . . 3 (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝑥𝐵 = 𝑥))
124, 11vtoclg 2790 . 2 (𝐶𝐴 → (𝐵𝐴 → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶)))
1312impcom 124 1 ((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  cop 3586   class class class wbr 3989   I cid 4273  cres 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-res 4623
This theorem is referenced by:  foeqcnvco  5769  f1eqcocnv  5770
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