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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvconstr | Structured version Visualization version GIF version |
Description: Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.) |
Ref | Expression |
---|---|
fvconstr.1 | ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) |
fvconstr.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
fvconstr.3 | ⊢ (𝜑 → 𝑌 ≠ ∅) |
Ref | Expression |
---|---|
fvconstr | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5170 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | fvconstr.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
3 | 2 | oveqd 7462 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
4 | df-ov 7448 | . . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
5 | 3, 4 | eqtrdi 2790 | . . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | fvconstr.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | fvconst2g 7237 | . . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) | |
9 | 7, 8 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) |
10 | 6, 9 | eqtrd 2774 | . . 3 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌) |
11 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌) | |
12 | fvconstr.3 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅) |
14 | 11, 13 | eqnetrd 3010 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅) |
15 | 5 | neeq1d 3002 | . . . . . 6 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
16 | 15 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
17 | 14, 16 | mpbid 232 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅) |
18 | dmxpss 6201 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
19 | ndmfv 6954 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
20 | 19 | necon1ai 2970 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
21 | 18, 20 | sselid 4000 | . . . 4 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅) |
22 | 17, 21 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 〈𝐴, 𝐵〉 ∈ 𝑅) |
23 | 10, 22 | impbida 800 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌)) |
24 | 1, 23 | bitrid 283 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∅c0 4347 {csn 4648 〈cop 4654 class class class wbr 5169 × cxp 5697 dom cdm 5699 ‘cfv 6572 (class class class)co 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 df-ov 7448 |
This theorem is referenced by: prsthinc 48639 prstchom2ALT 48664 |
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