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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfxrn2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.) |
| Ref | Expression |
|---|---|
| dfxrn2 | ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrnrel 38374 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
| 2 | dfrel4v 6210 | . . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}) | |
| 3 | 1, 2 | mpbi 230 | . 2 ⊢ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
| 4 | breq2 5147 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
| 5 | brxrn2 38376 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 6 | 5 | elv 3485 | . . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
| 7 | brxrn 38375 | . . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 8 | 7 | el3v 3488 | . . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
| 9 | 8 | anbi2i 623 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) |
| 10 | 3anass 1095 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
| 11 | 9, 10 | bitr4i 278 | . . . . . 6 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
| 12 | 11 | 2exbii 1849 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
| 13 | 4 | copsex2gb 5816 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
| 14 | 6, 12, 13 | 3bitr2i 299 | . . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
| 15 | 14 | simplbi 497 | . . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V)) |
| 16 | 4, 15 | cnvoprab 8085 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
| 17 | 8 | oprabbii 7500 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
| 18 | 17 | cnveqi 5885 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
| 19 | 3, 16, 18 | 3eqtr2i 2771 | 1 ⊢ (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 Vcvv 3480 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 ◡ccnv 5684 Rel wrel 5690 {coprab 7432 ⋉ 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-sbc 3789 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-oprab 7435 df-1st 8014 df-2nd 8015 df-xrn 38372 |
| This theorem is referenced by: (None) |
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