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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoels | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| dfcoels | ⊢ ∼ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-coels 38413 | . 2 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
| 2 | 1cossres 38430 | . 2 ⊢ ≀ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦)} | |
| 3 | brcnvep 38266 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)) | |
| 4 | 3 | elv 3485 | . . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢) | 
| 5 | brcnvep 38266 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢)) | |
| 6 | 5 | elv 3485 | . . . . 5 ⊢ (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢) | 
| 7 | 4, 6 | anbi12i 628 | . . . 4 ⊢ ((𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦) ↔ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)) | 
| 8 | 7 | rexbii 3094 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)) | 
| 9 | 8 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | 
| 10 | 1, 2, 9 | 3eqtri 2769 | 1 ⊢ ∼ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 {copab 5205 E cep 5583 ◡ccnv 5684 ↾ cres 5687 ≀ ccoss 38182 ∼ ccoels 38183 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 df-coss 38412 df-coels 38413 | 
| This theorem is referenced by: brcoels 38436 | 
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