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Mirrors > Home > MPE Home > Th. List > Mathboxes > coepr | Structured version Visualization version GIF version |
Description: Composition with the converse of epsilon. (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 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5227 | . . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴) |
4 | 1 | epelc 4951 | . . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
5 | 3, 4 | bitri 263 | . . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | anbi1i 727 | . . 3 ⊢ ((𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
7 | 6 | exbii 1764 | . 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) |
8 | coep.2 | . . 3 ⊢ 𝐵 ∈ V | |
9 | 1, 8 | brco 5214 | . 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵)) |
10 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)) | |
11 | 7, 9, 10 | 3bitr4i 291 | 1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 E cep 4947 ◡ccnv 5037 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-cnv 5046 df-co 5047 |
This theorem is referenced by: elfuns 31192 brub 31231 |
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