Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 36687 | . 2 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
2 | 1cossres 36704 | . 2 ⊢ ≀ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦)} | |
3 | brcnvep 36538 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)) | |
4 | 3 | elv 3447 | . . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢) |
5 | brcnvep 36538 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢)) | |
6 | 5 | elv 3447 | . . . . 5 ⊢ (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢) |
7 | 4, 6 | anbi12i 627 | . . . 4 ⊢ ((𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦) ↔ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)) |
8 | 7 | rexbii 3093 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)) |
9 | 8 | opabbii 5159 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢◡ E 𝑥 ∧ 𝑢◡ E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} |
10 | 1, 2, 9 | 3eqtri 2768 | 1 ⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1540 ∃wrex 3070 Vcvv 3441 class class class wbr 5092 {copab 5154 E cep 5523 ◡ccnv 5619 ↾ cres 5622 ≀ ccoss 36446 ∼ ccoels 36447 |
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-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-br 5093 df-opab 5155 df-eprel 5524 df-xp 5626 df-rel 5627 df-cnv 5628 df-res 5632 df-coss 36686 df-coels 36687 |
This theorem is referenced by: brcoels 36710 |
Copyright terms: Public domain | W3C validator |