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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 38355 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
2 | dfrel4v 6212 | . . 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 5152 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
5 | brxrn2 38357 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
6 | 5 | elv 3483 | . . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
7 | brxrn 38356 | . . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
8 | 7 | el3v 3486 | . . . . . . . 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 1094 | . . . . . . 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 1846 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
13 | 4 | copsex2gb 5819 | . . . . 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 8084 | . 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 5888 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
19 | 3, 16, 18 | 3eqtr2i 2769 | 1 ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 〈cop 4637 class class class wbr 5148 {copab 5210 × cxp 5687 ◡ccnv 5688 Rel wrel 5694 {coprab 7432 ⋉ cxrn 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 df-xrn 38353 |
This theorem is referenced by: (None) |
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