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Mirrors > Home > MPE Home > Th. List > Mathboxes > extep | Structured version Visualization version GIF version |
Description: Property of epsilon relation, see also extid 35562, extssr 35743 and the comment of df-ssr 35732. (Contributed by Peter Mazsa, 10-Jul-2019.) |
Ref | Expression |
---|---|
extep | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eccnvep 35533 | . 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) | |
2 | eccnvep 35533 | . 2 ⊢ (𝐵 ∈ 𝑊 → [𝐵]◡ E = 𝐵) | |
3 | 1, 2 | eqeqan12d 2838 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 E cep 5459 ◡ccnv 5549 [cec 8281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-eprel 5460 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ec 8285 |
This theorem is referenced by: (None) |
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