Theorem resieq 4829

Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

Assertion

Ref	Expression
resieq	⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))

Proof of Theorem resieq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3933	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶))
2		eqeq2 2149	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶))
3	1, 2	bibi12d 234	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
4	3	imbi2d 229	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))))
5		vex 2689	. . . . 5 ⊢ 𝑥 ∈ V
6	5	opres 4828	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7		df-br 3930	. . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
8	5	ideq 4691	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥)
9		df-br 3930	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
10	8, 9	bitr3i 185	. . . 4 ⊢ (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
11	6, 7, 10	3bitr4g 222	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥))
12	4, 11	vtoclg 2746	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
13	12	impcom 124	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929   I cid 4210   ↾ cres 4541

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-res 4551

This theorem is referenced by:  foeqcnvco  5691  f1eqcocnv  5692