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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvconstrn0 | Structured version Visualization version GIF version |
Description: Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
fvconstr.1 | ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) |
fvconstr.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
fvconstr.3 | ⊢ (𝜑 → 𝑌 ≠ ∅) |
Ref | Expression |
---|---|
fvconstrn0 | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5149 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | fvconstr.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
3 | 2 | oveqd 7448 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
4 | df-ov 7434 | . . . . . . 7 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
5 | 3, 4 | eqtrdi 2791 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | fvconstr.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | fvconst2g 7222 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) | |
9 | 7, 8 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) |
10 | 6, 9 | eqtrd 2775 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌) |
11 | fvconstr.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
12 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → 𝑌 ≠ ∅) |
13 | 10, 12 | eqnetrd 3006 | . . 3 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) ≠ ∅) |
14 | 5 | neeq1d 2998 | . . . . 5 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
15 | 14 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) ≠ ∅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅) |
16 | dmxpss 6193 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
17 | ndmfv 6942 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
18 | 17 | necon1ai 2966 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
19 | 16, 18 | sselid 3993 | . . . 4 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅) |
20 | 15, 19 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) ≠ ∅) → 〈𝐴, 𝐵〉 ∈ 𝑅) |
21 | 13, 20 | impbida 801 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
22 | 1, 21 | bitrid 283 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {csn 4631 〈cop 4637 class class class wbr 5148 × cxp 5687 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 |
This theorem is referenced by: prstchom 48878 prstchom2ALT 48880 |
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