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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 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5896 | . . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴) |
4 | 1 | epeli 5591 | . . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
5 | 3, 4 | bitri 275 | . . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | anbi1i 624 | . . 3 ⊢ ((𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
7 | 6 | exbii 1845 | . 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
8 | coep.2 | . . 3 ⊢ 𝐵 ∈ V | |
9 | 1, 8 | brco 5884 | . 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵)) |
10 | df-rex 3069 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) | |
11 | 7, 9, 10 | 3bitr4i 303 | 1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 class class class wbr 5148 E cep 5588 ◡ccnv 5688 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-cnv 5697 df-co 5698 |
This theorem is referenced by: elfuns 35897 brub 35936 |
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