Theorem dfres2 6030

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 5991	. 2 ⊢ Rel (𝑅 ↾ 𝐴)
2		relopabv 5794	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
3		vex 3458	. . . 4 ⊢ 𝑧 ∈ V
4		vex 3458	. . . 4 ⊢ 𝑤 ∈ V
5		eleq1w 2845	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
6		breq1 5103	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
7	5, 6	anbi12d 641	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦)))
8		breq2 5104	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
9	8	anbi2d 639	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
10	3, 4, 7, 9	opelopab 5513	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
11	4	brresi 5974	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
12		df-br 5101	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴))
13	10, 11, 12	3bitr2ri 302	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
14	1, 2, 13	eqrelriiv 5762	1 ⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ⟨cop 4588   class class class wbr 5100  {copab 5162   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-res 5659

This theorem is referenced by:  shftidt2  15094  bj-imdiridlem  37674  dfres4  38795  cnvepres  38800  ressn2  39028  tfsconcatrev  43922