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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxp2 | Structured version Visualization version GIF version |
Description: The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.) |
Ref | Expression |
---|---|
brtxp2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
brtxp2 | ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txpss3v 33339 | . . . . . . 7 ⊢ (𝑅 ⊗ 𝑆) ⊆ (V × (V × V)) | |
2 | 1 | brel 5617 | . . . . . 6 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V))) |
3 | 2 | simprd 498 | . . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → 𝐵 ∈ (V × V)) |
4 | elvv 5626 | . . . . 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 | 6, 7 | bitr4i 280 | . 2 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵)) |
9 | breq2 5070 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉)) | |
10 | 9 | pm5.32i 577 | . . 3 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉)) |
11 | 10 | 2exbii 1849 | . 2 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉)) |
12 | brtxp2.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
13 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
15 | 12, 13, 14 | brtxp 33341 | . . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉 ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) |
16 | 15 | anbi2i 624 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) |
17 | 3anass 1091 | . . . 4 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | |
18 | 16, 17 | bitr4i 280 | . . 3 ⊢ ((𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉) ↔ (𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) |
19 | 18 | 2exbii 1849 | . 2 ⊢ (∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴(𝑅 ⊗ 𝑆)〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) |
20 | 8, 11, 19 | 3bitri 299 | 1 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) |
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 × cxp 5553 ⊗ ctxp 33291 |
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-1st 7689 df-2nd 7690 df-txp 33315 |
This theorem is referenced by: brsuccf 33402 brrestrict 33410 |
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