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| Description: Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| seinxp | ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brinxp 5764 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | |
| 2 | 1 | ancoms 458 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 3 | 2 | rabbidva 3443 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) | 
| 4 | 3 | eleq1d 2826 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)) | 
| 5 | 4 | ralbiia 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V) | 
| 6 | df-se 5638 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
| 7 | df-se 5638 | . 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V) | |
| 8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 Se wse 5635 × cxp 5683 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-se 5638 df-xp 5691 | 
| This theorem is referenced by: (None) | 
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