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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 31161 | . . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V)) | |
2 | 1 | brel 5090 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → (〈𝑋, 𝑌〉 ∈ (V × V) ∧ 𝑍 ∈ (V × V))) |
3 | 2 | simprd 478 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V)) |
4 | elvv 5100 | . . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) | |
5 | 3, 4 | sylib 207 | . . . 4 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) |
6 | 5 | pm4.71ri 663 | . . 3 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
7 | 19.41vv 1902 | . . 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 266 | . 2 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
9 | breq2 4587 | . . . 4 ⊢ (𝑍 = 〈𝑧, 𝑤〉 → (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) | |
10 | 9 | pm5.32i 667 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
11 | 10 | 2exbii 1765 | . 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 3176 | . . . . . 6 ⊢ 𝑧 ∈ V | |
15 | vex 3176 | . . . . . 6 ⊢ 𝑤 ∈ V | |
16 | 12, 13, 14, 15 | brpprod 31162 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉 ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
17 | 16 | anbi2i 726 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) |
18 | 3anass 1035 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) | |
19 | 17, 18 | bitr4i 266 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
20 | 19 | 2exbii 1765 | . 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 285 | 1 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 × cxp 5036 pprodcpprod 31107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-pprod 31131 |
This theorem is referenced by: brpprod3b 31164 brapply 31215 dfrdg4 31228 |
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