![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > coepr | Structured version Visualization version GIF version |
Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
Ref | Expression |
---|---|
coep.1 | ⊢ 𝐴 ∈ V |
coep.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
coepr | ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coep.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | vex 3477 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5874 | . . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴) |
4 | 1 | epeli 5575 | . . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
5 | 3, 4 | bitri 274 | . . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | anbi1i 624 | . . 3 ⊢ ((𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
7 | 6 | exbii 1850 | . 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
8 | coep.2 | . . 3 ⊢ 𝐵 ∈ V | |
9 | 1, 8 | brco 5862 | . 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵)) |
10 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) | |
11 | 7, 9, 10 | 3bitr4i 302 | 1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ∃wrex 3069 Vcvv 3473 class class class wbr 5141 E cep 5572 ◡ccnv 5668 ∘ ccom 5673 |
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 5292 ax-nul 5299 ax-pr 5420 |
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-ne 2940 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-eprel 5573 df-cnv 5677 df-co 5678 |
This theorem is referenced by: elfuns 34717 brub 34756 |
Copyright terms: Public domain | W3C validator |