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Mirrors > Home > MPE Home > Th. List > Mathboxes > coep | Structured version Visualization version GIF version |
Description: Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
Ref | Expression |
---|---|
coep.1 | ⊢ 𝐴 ∈ V |
coep.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
coep | ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coep.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
2 | 1 | epeli 5432 | . . . 4 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | anbi1ci 628 | . . 3 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
4 | 3 | exbii 1849 | . 2 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
5 | coep.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5, 1 | brco 5705 | . 2 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵)) |
7 | df-rex 3112 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) | |
8 | 4, 6, 7 | 3bitr4i 306 | 1 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 E cep 5429 ∘ ccom 5523 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-co 5528 |
This theorem is referenced by: dffr5 33102 brbigcup 33472 elfuns 33489 brimage 33500 |
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