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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 5496 | . . . 4 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | anbi1ci 625 | . . 3 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
4 | 3 | exbii 1853 | . 2 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
5 | coep.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5, 1 | brco 5776 | . 2 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵)) |
7 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) | |
8 | 4, 6, 7 | 3bitr4i 302 | 1 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1785 ∈ wcel 2109 ∃wrex 3066 Vcvv 3430 class class class wbr 5078 E cep 5493 ∘ ccom 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ne 2945 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-eprel 5494 df-co 5597 |
This theorem is referenced by: dffr5 33700 brbigcup 34179 elfuns 34196 brimage 34207 |
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