Theorem resieq 4910

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 4002	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵( I ↾ 𝐴)𝐶))
2		eqeq2 2185	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐶))
3	1, 2	bibi12d 235	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥) ↔ (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
4	3	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))))
5		vex 2738	. . . . 5 ⊢ 𝑥 ∈ V
6	5	opres 4909	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ ⟨𝐵, 𝑥⟩ ∈ I ))
7		df-br 3999	. . . 4 ⊢ (𝐵( I ↾ 𝐴)𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ ( I ↾ 𝐴))
8	5	ideq 4772	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ 𝐵 = 𝑥)
9		df-br 3999	. . . . 5 ⊢ (𝐵 I 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
10	8, 9	bitr3i 186	. . . 4 ⊢ (𝐵 = 𝑥 ↔ ⟨𝐵, 𝑥⟩ ∈ I )
11	6, 7, 10	3bitr4g 223	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝑥 ↔ 𝐵 = 𝑥))
12	4, 11	vtoclg 2795	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)))
13	12	impcom 125	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ⟨cop 3592   class class class wbr 3998   I cid 4282   ↾ cres 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-res 4632

This theorem is referenced by:  foeqcnvco  5781  f1eqcocnv  5782