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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnxrnidres | Structured version Visualization version GIF version |
Description: Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
Ref | Expression |
---|---|
rnxrnidres | ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnxrnres 35649 | . 2 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)} | |
2 | ideqg 5724 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑢 I 𝑦 ↔ 𝑢 = 𝑦)) | |
3 | 2 | elv 3501 | . . . . 5 ⊢ (𝑢 I 𝑦 ↔ 𝑢 = 𝑦) |
4 | 3 | anbi1ci 627 | . . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)) |
5 | 4 | rexbii 3249 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)) |
6 | 5 | opabbii 5135 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)} |
7 | 1, 6 | eqtri 2846 | 1 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wrex 3141 Vcvv 3496 class class class wbr 5068 {copab 5130 I cid 5461 ran crn 5558 ↾ cres 5559 ⋉ cxrn 35454 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-ec 8293 df-xrn 35625 |
This theorem is referenced by: (None) |
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