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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 35784 | . . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5581 | . . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 499 | . . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5590 | . . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | sylib 221 | . . . 4 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) |
6 | 5 | pm4.71ri 564 | . . 3 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵)) |
7 | 19.41vv 1951 | . . 3 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵)) | |
8 | breq2 5034 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
9 | 8 | pm5.32i 578 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
10 | 9 | 2exbii 1850 | . . 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 302 | . 2 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
12 | brxrn 35786 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
13 | 12 | el3v23 35657 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
14 | 13 | anbi2d 631 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))) |
15 | 3anass 1092 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
16 | 14, 15 | syl6bbr 292 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
17 | 16 | 2exbidv 1925 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
18 | 11, 17 | syl5bb 286 | 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 × cxp 5517 ⋉ cxrn 35612 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-xrn 35783 |
This theorem is referenced by: dfxrn2 35788 elecxrn 35798 inxpxrn 35803 br1cnvxrn2 35804 |
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