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Mirrors > Home > MPE Home > Th. List > eqbrrdva | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation 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 5705 | . . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V) | |
3 | 1, 2 | sstrdi 4008 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V)) |
4 | df-rel 5696 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
5 | 3, 4 | sylibr 234 | . 2 ⊢ (𝜑 → Rel 𝐴) |
6 | eqbrrdva.2 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷)) | |
7 | 6, 2 | sstrdi 4008 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V)) |
8 | df-rel 5696 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
9 | 7, 8 | sylibr 234 | . 2 ⊢ (𝜑 → Rel 𝐵) |
10 | 1 | ssbrd 5191 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
11 | brxp 5738 | . . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) | |
12 | 10, 11 | imbitrdi 251 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
13 | 6 | ssbrd 5191 | . . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
14 | 13, 11 | imbitrdi 251 | . . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
15 | eqbrrdva.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
16 | 15 | 3expib 1121 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
17 | 12, 14, 16 | pm5.21ndd 379 | . 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
18 | 5, 9, 17 | eqbrrdv 5806 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 × cxp 5687 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: metustsym 24584 |
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