Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rnxrncnvepres | Structured version Visualization version GIF version |
Description: Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
Ref | Expression |
---|---|
rnxrncnvepres | ⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnxrnres 36513 | . 2 ⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦)} | |
2 | brcnvep 36392 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢)) | |
3 | 2 | elv 3437 | . . . . 5 ⊢ (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢) |
4 | 3 | anbi1ci 626 | . . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦) ↔ (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)) |
5 | 4 | rexbii 3180 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)) |
6 | 5 | opabbii 5146 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)} |
7 | 1, 6 | eqtri 2768 | 1 ⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 Vcvv 3431 class class class wbr 5079 {copab 5141 E cep 5494 ◡ccnv 5588 ran crn 5590 ↾ cres 5591 ⋉ cxrn 36320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-eprel 5495 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-fo 6437 df-fv 6439 df-1st 7818 df-2nd 7819 df-ec 8475 df-xrn 36489 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |