Theorem resieq 5989

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 5113	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶))
2		eqeq2 2775	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶))
3	1, 2	bibi12d 348	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
4	3	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))))
5		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
6	5	opres 5988	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7		df-br 5110	. . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
8	5	ideq 5838	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥)
9		df-br 5110	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
10	8, 9	bitr3i 280	. . . 4 ⊢ (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
11	6, 7, 10	3bitr4g 317	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥))
12	4, 11	vtoclg 3522	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
13	12	impcom 412	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109   I cid 5555   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by:  foeqcnvco  7298  f1eqcocnv  7299  dfle2  13167  pospo  18394  dirref  18652  ustref  24376  trust  24386  brfvrcld2  44438  resipos  49773