|   | 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 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | 1, 2 | brcnv 5893 | . . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴) | 
| 4 | 1 | epeli 5586 | . . . . 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 1848 | . 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) | 
| 8 | coep.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 9 | 1, 8 | brco 5881 | . 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵)) | 
| 10 | df-rex 3071 | . 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 1779 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 E cep 5583 ◡ccnv 5684 ∘ ccom 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-cnv 5693 df-co 5694 | 
| This theorem is referenced by: elfuns 35916 brub 35955 | 
| Copyright terms: Public domain | W3C validator |