Theorem dfres2 6043

Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Assertion

Ref	Expression
dfres2	⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dfres2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 6004	. 2 ⊢ Rel (𝑅 ↾ 𝐴)
2		relopabv 5808	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
3		vex 3459	. . . 4 ⊢ 𝑧 ∈ V
4		vex 3459	. . . 4 ⊢ 𝑤 ∈ V
5		eleq1w 2846	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
6		breq1 5112	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
7	5, 6	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦)))
8		breq2 5113	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
9	8	anbi2d 641	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
10	3, 4, 7, 9	opelopab 5527	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
11	4	brresi 5987	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
12		df-br 5110	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴))
13	10, 11, 12	3bitr2ri 303	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
14	1, 2, 13	eqrelriiv 5776	1 ⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109  {copab 5173   ↾ 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-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by:  shftidt2  15114  bj-imdiridlem  37829  dfres4  38948  cnvepres  38953  ressn2  39181  tfsconcatrev  44075