Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > brresg | Structured version Visualization version GIF version |
Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) |
Ref | Expression |
---|---|
brresg | ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelresg 5324 | . 2 ⊢ (𝐵 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) | |
2 | df-br 4584 | . 2 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷)) | |
3 | df-br 4584 | . . 3 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
4 | 3 | anbi1i 727 | . 2 ⊢ ((𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
5 | 1, 2, 4 | 3bitr4g 302 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 ↾ cres 5040 |
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-res 5050 |
This theorem is referenced by: fv1stcnv 30925 fv2ndcnv 30926 |
Copyright terms: Public domain | W3C validator |