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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'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 5149 | . . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V) | |
3 | 1, 2 | syl6ss 3580 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V)) |
4 | df-rel 5045 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
5 | 3, 4 | sylibr 223 | . 2 ⊢ (𝜑 → Rel 𝐴) |
6 | eqbrrdva.2 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷)) | |
7 | 6, 2 | syl6ss 3580 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V)) |
8 | df-rel 5045 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
9 | 7, 8 | sylibr 223 | . 2 ⊢ (𝜑 → Rel 𝐵) |
10 | 1 | ssbrd 4626 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
11 | brxp 5071 | . . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) | |
12 | 10, 11 | syl6ib 240 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
13 | 6 | ssbrd 4626 | . . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
14 | 13, 11 | syl6ib 240 | . . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
15 | eqbrrdva.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
16 | 15 | 3expib 1260 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
17 | 12, 14, 16 | pm5.21ndd 368 | . 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
18 | 5, 9, 17 | eqbrrdv 5140 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: metustsym 22170 |
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