Theorem seinxp 5757

Description: Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
seinxp	⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Proof of Theorem seinxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brinxp 5752	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2	1	ancoms 457	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	2	rabbidva 3426	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4	3	eleq1d 2811	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
5	4	ralbiia 3081	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6		df-se 5630	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
7		df-se 5630	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
8	5, 6, 7	3bitr4i 302	1 ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2099  ∀wral 3051  {crab 3419  Vcvv 3462   ∩ cin 3945   class class class wbr 5145   Se wse 5627   × cxp 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-se 5630  df-xp 5680

This theorem is referenced by: (None)