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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 36149 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
2 | dfrel4v 6023 | . . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}) | |
3 | 1, 2 | mpbi 233 | . 2 ⊢ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
4 | breq2 5035 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
5 | brxrn2 36151 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
6 | 5 | elv 3405 | . . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
7 | brxrn 36150 | . . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
8 | 7 | el3v 36017 | . . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
9 | 8 | anbi2i 626 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) |
10 | 3anass 1096 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
11 | 9, 10 | bitr4i 281 | . . . . . 6 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
12 | 11 | 2exbii 1855 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
13 | 4 | copsex2gb 5651 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
14 | 6, 12, 13 | 3bitr2i 302 | . . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
15 | 14 | simplbi 501 | . . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V)) |
16 | 4, 15 | cnvoprab 7786 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
17 | 8 | oprabbii 7238 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
18 | 17 | cnveqi 5718 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
19 | 3, 16, 18 | 3eqtr2i 2768 | 1 ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∃wex 1786 ∈ wcel 2114 Vcvv 3399 〈cop 4523 class class class wbr 5031 {copab 5093 × cxp 5524 ◡ccnv 5525 Rel wrel 5531 {coprab 7174 ⋉ cxrn 35978 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fo 6346 df-fv 6348 df-oprab 7177 df-1st 7717 df-2nd 7718 df-xrn 36147 |
This theorem is referenced by: (None) |
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