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Mirrors > Home > MPE Home > Th. List > Mathboxes > coep | Structured version Visualization version GIF version |
Description: Composition with epsilon. (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 | . . . . . 6 ⊢ 𝐵 ∈ V | |
2 | 1 | epeli 5257 | . . . . 5 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | anbi2i 618 | . . . 4 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ (𝐴𝑅𝑥 ∧ 𝑥 ∈ 𝐵)) |
4 | ancom 454 | . . . 4 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) | |
5 | 3, 4 | bitri 267 | . . 3 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
6 | 5 | exbii 1949 | . 2 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) |
7 | coep.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | 7, 1 | brco 5525 | . 2 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵)) |
9 | df-rex 3123 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥)) | |
10 | 6, 8, 9 | 3bitr4i 295 | 1 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∃wex 1880 ∈ wcel 2166 ∃wrex 3118 Vcvv 3414 class class class wbr 4873 E cep 5254 ∘ ccom 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-eprel 5255 df-co 5351 |
This theorem is referenced by: dffr5 32185 brbigcup 32544 elfuns 32561 brimage 32572 |
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