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Mirrors > Home > MPE Home > Th. List > Mathboxes > brpprod3a | Structured version Visualization version GIF version |
Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brpprod3.1 | ⊢ 𝑋 ∈ V |
brpprod3.2 | ⊢ 𝑌 ∈ V |
brpprod3.3 | ⊢ 𝑍 ∈ V |
Ref | Expression |
---|---|
brpprod3a | ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pprodss4v 33345 | . . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V)) | |
2 | 1 | brel 5617 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → (〈𝑋, 𝑌〉 ∈ (V × V) ∧ 𝑍 ∈ (V × V))) |
3 | 2 | simprd 498 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V)) |
4 | elvv 5626 | . . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) | |
5 | 3, 4 | sylib 220 | . . . 4 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) |
6 | 5 | pm4.71ri 563 | . . 3 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
7 | 19.41vv 1951 | . . 3 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) | |
8 | 6, 7 | bitr4i 280 | . 2 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
9 | breq2 5070 | . . . 4 ⊢ (𝑍 = 〈𝑧, 𝑤〉 → (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) | |
10 | 9 | pm5.32i 577 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
11 | 10 | 2exbii 1849 | . 2 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
12 | brpprod3.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
13 | brpprod3.2 | . . . . . 6 ⊢ 𝑌 ∈ V | |
14 | vex 3497 | . . . . . 6 ⊢ 𝑧 ∈ V | |
15 | vex 3497 | . . . . . 6 ⊢ 𝑤 ∈ V | |
16 | 12, 13, 14, 15 | brpprod 33346 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉 ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
17 | 16 | anbi2i 624 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) |
18 | 3anass 1091 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) | |
19 | 17, 18 | bitr4i 280 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
20 | 19 | 2exbii 1849 | . 2 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
21 | 8, 11, 20 | 3bitri 299 | 1 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
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 pprodcpprod 33292 |
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 df-pprod 33316 |
This theorem is referenced by: brpprod3b 33348 brapply 33399 dfrdg4 33412 |
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