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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brxrn2 | Structured version Visualization version GIF version |
Description: A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.) |
Ref | Expression |
---|---|
brxrn2 | ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnss3v 36772 | . . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5695 | . . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 496 | . . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5704 | . . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | sylib 217 | . . . 4 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) |
6 | 5 | pm4.71ri 561 | . . 3 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵)) |
7 | 19.41vv 1954 | . . 3 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵)) | |
8 | breq2 5107 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
9 | 8 | pm5.32i 575 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
10 | 9 | 2exbii 1851 | . . 3 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
11 | 6, 7, 10 | 3bitr2i 298 | . 2 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
12 | brxrn 36774 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
13 | 12 | el3v23 36621 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
14 | 13 | anbi2d 629 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))) |
15 | 3anass 1095 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
16 | 14, 15 | bitr4di 288 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
17 | 16 | 2exbidv 1927 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
18 | 11, 17 | bitrid 282 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 〈cop 4590 class class class wbr 5103 × cxp 5629 ⋉ cxrn 36571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fo 6499 df-fv 6501 df-1st 7913 df-2nd 7914 df-xrn 36771 |
This theorem is referenced by: dfxrn2 36776 elecxrn 36786 inxpxrn 36795 br1cnvxrn2 36796 |
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