Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > brcnvepres | Structured version Visualization version GIF version |
Description: Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.) |
Ref | Expression |
---|---|
brcnvepres | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brres 5832 | . 2 ⊢ (𝐶 ∈ 𝑊 → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝐶))) | |
2 | brcnvep 36027 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝐶 ↔ 𝐶 ∈ 𝐵)) | |
3 | 2 | anbi2d 632 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))) |
4 | 1, 3 | sylan9bbr 514 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 class class class wbr 5030 E cep 5433 ◡ccnv 5524 ↾ cres 5527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rex 3059 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-eprel 5434 df-xp 5531 df-rel 5532 df-cnv 5533 df-res 5537 |
This theorem is referenced by: dfeldisj3 36453 dfeldisj4 36454 |
Copyright terms: Public domain | W3C validator |