Theorem resieq 5991

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 5115	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶))
2		eqeq2 2777	. . . . 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 3461	. . . . 5 ⊢ 𝑥 ∈ V
6	5	opres 5990	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7		df-br 5112	. . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
8	5	ideq 5840	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥)
9		df-br 5112	. . . . 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 3524	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
13	12	impcom 413	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   class class class wbr 5111   I cid 5557   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-res 5675

This theorem is used by:  foeqcnvco  7307  f1eqcocnv  7308  dfle2  13190  pospo  18423  dirref  18681  ustref  24429  trust  24439  brfvrcld2  44478  resipos  49812