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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 35626 | . . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5619 | . . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 498 | . . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5628 | . . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | sylib 220 | . . . 4 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) |
6 | 5 | pm4.71ri 563 | . . 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 5072 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
9 | 8 | pm5.32i 577 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
10 | 9 | 2exbii 1849 | . . 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 301 | . 2 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) |
12 | brxrn 35628 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
13 | 12 | el3v23 35499 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
14 | 13 | anbi2d 630 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))) |
15 | 3anass 1091 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
16 | 14, 15 | syl6bbr 291 | . . 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 285 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 〈cop 4575 class class class wbr 5068 × cxp 5555 ⋉ cxrn 35454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-xrn 35625 |
This theorem is referenced by: dfxrn2 35630 elecxrn 35640 inxpxrn 35645 br1cnvxrn2 35646 |
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