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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 35618 | . . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5612 | . . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 498 | . . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5621 | . . . . 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 1947 | . . 3 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵)) | |
8 | breq2 5063 | . . . . 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 1845 | . . 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 35620 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
13 | 12 | el3v23 35491 | . . . . 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 1921 | . 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 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 〈cop 4567 class class class wbr 5059 × cxp 5548 ⋉ cxrn 35446 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fo 6356 df-fv 6358 df-1st 7683 df-2nd 7684 df-xrn 35617 |
This theorem is referenced by: dfxrn2 35622 elecxrn 35632 inxpxrn 35637 br1cnvxrn2 35638 |
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