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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 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5753 | . . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴) |
4 | 1 | epeli 5468 | . . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
5 | 3, 4 | bitri 277 | . . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | anbi1i 625 | . . 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 5741 | . 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵)) |
10 | df-rex 3144 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) | |
11 | 7, 9, 10 | 3bitr4i 305 | 1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 E cep 5464 ◡ccnv 5554 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-cnv 5563 df-co 5564 |
This theorem is referenced by: elfuns 33376 brub 33415 |
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