![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eqbrrdiv | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqbrrdiv.1 | ⊢ Rel 𝐴 |
eqbrrdiv.2 | ⊢ Rel 𝐵 |
eqbrrdiv.3 | ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdiv | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdiv.1 | . 2 ⊢ Rel 𝐴 | |
2 | eqbrrdiv.2 | . 2 ⊢ Rel 𝐵 | |
3 | eqbrrdiv.3 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
4 | df-br 5111 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
5 | df-br 5111 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
7 | 1, 2, 6 | eqrelrdv 5753 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ⟨cop 4597 class class class wbr 5110 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 |
This theorem is referenced by: eqfunresadj 7310 funcpropd 17794 fullpropd 17814 fthpropd 17815 dvres 25291 |
Copyright terms: Public domain | W3C validator |