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Mirrors > Home > ILE Home > Th. List > seinxp | GIF version |
Description: Intersection of set-like relation with cross 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 4351 | . . . . . 6 ⊢ ((y ∈ A ∧ x ∈ A) → (y𝑅x ↔ y(𝑅 ∩ (A × A))x)) | |
2 | 1 | ancoms 255 | . . . . 5 ⊢ ((x ∈ A ∧ y ∈ A) → (y𝑅x ↔ y(𝑅 ∩ (A × A))x)) |
3 | 2 | rabbidva 2542 | . . . 4 ⊢ (x ∈ A → {y ∈ A ∣ y𝑅x} = {y ∈ A ∣ y(𝑅 ∩ (A × A))x}) |
4 | 3 | eleq1d 2103 | . . 3 ⊢ (x ∈ A → ({y ∈ A ∣ y𝑅x} ∈ V ↔ {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V)) |
5 | 4 | ralbiia 2332 | . 2 ⊢ (∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V ↔ ∀x ∈ A {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V) |
6 | df-se 4056 | . 2 ⊢ (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V) | |
7 | df-se 4056 | . 2 ⊢ ((𝑅 ∩ (A × A)) Se A ↔ ∀x ∈ A {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V) | |
8 | 5, 6, 7 | 3bitr4i 201 | 1 ⊢ (𝑅 Se A ↔ (𝑅 ∩ (A × A)) Se A) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1390 ∀wral 2300 {crab 2304 Vcvv 2551 ∩ cin 2910 class class class wbr 3755 Se wse 4055 × cxp 4286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-se 4056 df-xp 4294 |
This theorem is referenced by: (None) |
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