Theorem dfres2 6034

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 6005	. 2 ⊢ Rel (𝑅 ↾ 𝐴)
2		relopabv 5816	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
3		vex 3479	. . . 4 ⊢ 𝑧 ∈ V
4		vex 3479	. . . 4 ⊢ 𝑤 ∈ V
5		eleq1w 2817	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
6		breq1 5147	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
7	5, 6	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦)))
8		breq2 5148	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
9	8	anbi2d 630	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
10	3, 4, 7, 9	opelopab 5538	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
11	4	brresi 5985	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
12		df-br 5145	. . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴))
13	10, 11, 12	3bitr2ri 300	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑅 ↾ 𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
14	1, 2, 13	eqrelriiv 5785	1 ⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ⟨cop 4630   class class class wbr 5144  {copab 5206   ↾ cres 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-res 5684

This theorem is referenced by:  shftidt2  15015  bj-imdiridlem  35971  dfres4  37068  cnvepres  37073  ressn2  37218  tfsconcatrev  41969