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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfxrn2 | Structured version Visualization version GIF version |
Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.) |
Ref | Expression |
---|---|
dfxrn2 | ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnrel 37706 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
2 | dfrel4v 6189 | . . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}) | |
3 | 1, 2 | mpbi 229 | . 2 ⊢ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
4 | breq2 5152 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
5 | brxrn2 37708 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
6 | 5 | elv 3479 | . . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
7 | brxrn 37707 | . . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
8 | 7 | el3v 37549 | . . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
9 | 8 | anbi2i 622 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) |
10 | 3anass 1094 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
11 | 9, 10 | bitr4i 278 | . . . . . 6 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
12 | 11 | 2exbii 1850 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
13 | 4 | copsex2gb 5806 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
14 | 6, 12, 13 | 3bitr2i 299 | . . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
15 | 14 | simplbi 497 | . . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V)) |
16 | 4, 15 | cnvoprab 8050 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
17 | 8 | oprabbii 7479 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
18 | 17 | cnveqi 5874 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
19 | 3, 16, 18 | 3eqtr2i 2765 | 1 ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 〈cop 4634 class class class wbr 5148 {copab 5210 × cxp 5674 ◡ccnv 5675 Rel wrel 5681 {coprab 7413 ⋉ cxrn 37505 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-oprab 7416 df-1st 7979 df-2nd 7980 df-xrn 37704 |
This theorem is referenced by: (None) |
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