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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cnvres | Structured version Visualization version GIF version |
Description: Binary relation on the converse of a restriction. (Contributed by Peter Mazsa, 27-Jul-2019.) |
Ref | Expression |
---|---|
br1cnvres | ⊢ (𝐵 ∈ 𝑉 → (𝐵◡(𝑅 ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5680 | . . . 4 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
2 | 1 | cnveqi 5865 | . . 3 ⊢ ◡(𝑅 ↾ 𝐴) = ◡(𝑅 ∩ (𝐴 × V)) |
3 | 2 | breqi 5146 | . 2 ⊢ (𝐵◡(𝑅 ↾ 𝐴)𝐶 ↔ 𝐵◡(𝑅 ∩ (𝐴 × V))𝐶) |
4 | elex 3490 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
5 | br1cnvinxp 36915 | . . . . 5 ⊢ (𝐵◡(𝑅 ∩ (𝐴 × V))𝐶 ↔ ((𝐵 ∈ V ∧ 𝐶 ∈ 𝐴) ∧ 𝐶𝑅𝐵)) | |
6 | anass 469 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐶 ∈ 𝐴) ∧ 𝐶𝑅𝐵) ↔ (𝐵 ∈ V ∧ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) | |
7 | 5, 6 | bitri 274 | . . . 4 ⊢ (𝐵◡(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐵 ∈ V ∧ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) |
8 | 7 | baib 536 | . . 3 ⊢ (𝐵 ∈ V → (𝐵◡(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) |
10 | 3, 9 | bitrid 282 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡(𝑅 ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3472 ∩ cin 3942 class class class wbr 5140 × cxp 5666 ◡ccnv 5667 ↾ cres 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5291 ax-nul 5298 ax-pr 5419 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3474 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5141 df-opab 5203 df-xp 5674 df-rel 5675 df-cnv 5676 df-res 5680 |
This theorem is referenced by: coss1cnvres 37078 |
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