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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rnxrn | Structured version Visualization version GIF version | ||
| Description: Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.) | 
| Ref | Expression | 
|---|---|
| rnxrn | ⊢ ran (𝑅 ⋉ 𝑆) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 3anass 1095 | . . . . 5 ⊢ ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 2 | 1 | 3exbii 1850 | . . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | 
| 3 | exrot3 2165 | . . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦∃𝑢(𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 4 | 19.42v 1953 | . . . . 5 ⊢ (∃𝑢(𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 5 | 4 | 2exbii 1849 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑢(𝑤 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | 
| 6 | 2, 3, 5 | 3bitri 297 | . . 3 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | 
| 7 | 6 | abbii 2809 | . 2 ⊢ {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))} | 
| 8 | dfrn6 38303 | . . 3 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅} | |
| 9 | n0 4353 | . . . . 5 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆)) | |
| 10 | elec1cnvxrn2 38398 | . . . . . . 7 ⊢ (𝑢 ∈ V → (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 11 | 10 | elv 3485 | . . . . . 6 ⊢ (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) | 
| 12 | 11 | exbii 1848 | . . . . 5 ⊢ (∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) | 
| 13 | 9, 12 | bitri 275 | . . . 4 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) | 
| 14 | 13 | abbii 2809 | . . 3 ⊢ {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | 
| 15 | 8, 14 | eqtri 2765 | . 2 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | 
| 16 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))} | |
| 17 | 7, 15, 16 | 3eqtr4i 2775 | 1 ⊢ ran (𝑅 ⋉ 𝑆) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ≠ wne 2940 Vcvv 3480 ∅c0 4333 〈cop 4632 class class class wbr 5143 {copab 5205 ◡ccnv 5684 ran crn 5686 [cec 8743 ⋉ cxrn 38181 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-ec 8747 df-xrn 38372 | 
| This theorem is referenced by: rnxrnres 38400 dfcoss4 38416 dfssr2 38500 | 
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