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Mirrors > Home > ILE Home > Th. List > brab2ga | GIF version |
Description: The law of concretion for a binary relation. See brab2a 4336 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
brab2ga.1 | ⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) |
brab2ga.2 | ⊢ 𝑅 = {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} |
Ref | Expression |
---|---|
brab2ga | ⊢ (A𝑅B ↔ ((A ∈ 𝐶 ∧ B ∈ 𝐷) ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brab2ga.2 | . . . 4 ⊢ 𝑅 = {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} | |
2 | opabssxp 4357 | . . . 4 ⊢ {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} ⊆ (𝐶 × 𝐷) | |
3 | 1, 2 | eqsstri 2969 | . . 3 ⊢ 𝑅 ⊆ (𝐶 × 𝐷) |
4 | 3 | brel 4335 | . 2 ⊢ (A𝑅B → (A ∈ 𝐶 ∧ B ∈ 𝐷)) |
5 | df-br 3756 | . . . 4 ⊢ (A𝑅B ↔ 〈A, B〉 ∈ 𝑅) | |
6 | 1 | eleq2i 2101 | . . . 4 ⊢ (〈A, B〉 ∈ 𝑅 ↔ 〈A, B〉 ∈ {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)}) |
7 | 5, 6 | bitri 173 | . . 3 ⊢ (A𝑅B ↔ 〈A, B〉 ∈ {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)}) |
8 | brab2ga.1 | . . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) | |
9 | 8 | opelopab2a 3993 | . . 3 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (〈A, B〉 ∈ {〈x, y〉 ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} ↔ ψ)) |
10 | 7, 9 | syl5bb 181 | . 2 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (A𝑅B ↔ ψ)) |
11 | 4, 10 | biadan2 429 | 1 ⊢ (A𝑅B ↔ ((A ∈ 𝐶 ∧ B ∈ 𝐷) ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 〈cop 3370 class class class wbr 3755 {copab 3808 × 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-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 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-xp 4294 |
This theorem is referenced by: reapval 7360 ltxr 8465 |
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