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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 37483, extssr 37683 and the comment of df-ssr 37672. (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 37454 | . 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) | |
2 | eccnvep 37454 | . 2 ⊢ (𝐵 ∈ 𝑊 → [𝐵]◡ E = 𝐵) | |
3 | 1, 2 | eqeqan12d 2745 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 E cep 5579 ◡ccnv 5675 [cec 8705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ec 8709 |
This theorem is referenced by: (None) |
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