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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 37844 | . . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5743 | . . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 495 | . . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5752 | . . . . 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 560 | . . 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 5152 | . . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩)) | |
9 | 8 | pm5.32i 574 | . . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩)) |
10 | 9 | 2exbii 1844 | . . 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 299 | . 2 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩)) |
12 | brxrn 37846 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
13 | 12 | el3v23 37697 | . . . . 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 1093 | . . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
16 | 14, 15 | bitr4di 289 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
17 | 16 | 2exbidv 1920 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
18 | 11, 17 | bitrid 283 | 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 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 class class class wbr 5148 × cxp 5676 ⋉ cxrn 37647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-1st 7993 df-2nd 7994 df-xrn 37843 |
This theorem is referenced by: dfxrn2 37848 elecxrn 37858 inxpxrn 37867 br1cnvxrn2 37868 |
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