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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 35640 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
2 | dfrel4v 6047 | . . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}) | |
3 | 1, 2 | mpbi 232 | . 2 ⊢ (𝑅 ⋉ 𝑆) = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
4 | breq2 5070 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉)) | |
5 | brxrn2 35642 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
6 | 5 | elv 3499 | . . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
7 | brxrn 35641 | . . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
8 | 7 | el3v 35506 | . . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉 ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
9 | 8 | anbi2i 624 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) |
10 | 3anass 1091 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))) | |
11 | 9, 10 | bitr4i 280 | . . . . . 6 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
12 | 11 | 2exbii 1849 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) |
13 | 4 | copsex2gb 5679 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
14 | 6, 12, 13 | 3bitr2i 301 | . . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧)) |
15 | 14 | simplbi 500 | . . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V)) |
16 | 4, 15 | cnvoprab 7758 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈𝑢, 𝑧〉 ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧} |
17 | 8 | oprabbii 7221 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
18 | 17 | cnveqi 5745 | . 2 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ 𝑢(𝑅 ⋉ 𝑆)〈𝑥, 𝑦〉} = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
19 | 3, 16, 18 | 3eqtr2i 2850 | 1 ⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 {copab 5128 × cxp 5553 ◡ccnv 5554 Rel wrel 5560 {coprab 7157 ⋉ cxrn 35467 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-oprab 7160 df-1st 7689 df-2nd 7690 df-xrn 35638 |
This theorem is referenced by: (None) |
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