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Mirrors > Home > ILE Home > Th. List > eqbrrdva | GIF version |
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.) |
Ref | Expression |
---|---|
eqbrrdva.1 | ⊢ (φ → A ⊆ (𝐶 × 𝐷)) |
eqbrrdva.2 | ⊢ (φ → B ⊆ (𝐶 × 𝐷)) |
eqbrrdva.3 | ⊢ ((φ ∧ x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy)) |
Ref | Expression |
---|---|
eqbrrdva | ⊢ (φ → A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdva.1 | . . . 4 ⊢ (φ → A ⊆ (𝐶 × 𝐷)) | |
2 | xpss 4389 | . . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V) | |
3 | 1, 2 | syl6ss 2951 | . . 3 ⊢ (φ → A ⊆ (V × V)) |
4 | df-rel 4295 | . . 3 ⊢ (Rel A ↔ A ⊆ (V × V)) | |
5 | 3, 4 | sylibr 137 | . 2 ⊢ (φ → Rel A) |
6 | eqbrrdva.2 | . . . 4 ⊢ (φ → B ⊆ (𝐶 × 𝐷)) | |
7 | 6, 2 | syl6ss 2951 | . . 3 ⊢ (φ → B ⊆ (V × V)) |
8 | df-rel 4295 | . . 3 ⊢ (Rel B ↔ B ⊆ (V × V)) | |
9 | 7, 8 | sylibr 137 | . 2 ⊢ (φ → Rel B) |
10 | 1 | ssbrd 3796 | . . . 4 ⊢ (φ → (xAy → x(𝐶 × 𝐷)y)) |
11 | brxp 4318 | . . . 4 ⊢ (x(𝐶 × 𝐷)y ↔ (x ∈ 𝐶 ∧ y ∈ 𝐷)) | |
12 | 10, 11 | syl6ib 150 | . . 3 ⊢ (φ → (xAy → (x ∈ 𝐶 ∧ y ∈ 𝐷))) |
13 | 6 | ssbrd 3796 | . . . 4 ⊢ (φ → (xBy → x(𝐶 × 𝐷)y)) |
14 | 13, 11 | syl6ib 150 | . . 3 ⊢ (φ → (xBy → (x ∈ 𝐶 ∧ y ∈ 𝐷))) |
15 | eqbrrdva.3 | . . . 4 ⊢ ((φ ∧ x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy)) | |
16 | 15 | 3expib 1106 | . . 3 ⊢ (φ → ((x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy))) |
17 | 12, 14, 16 | pm5.21ndd 620 | . 2 ⊢ (φ → (xAy ↔ xBy)) |
18 | 5, 9, 17 | eqbrrdv 4380 | 1 ⊢ (φ → A = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 class class class wbr 3755 × cxp 4286 Rel wrel 4293 |
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-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 df-rel 4295 |
This theorem is referenced by: (None) |
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