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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 5040 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | fvconstr.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
3 | 2 | oveqd 7208 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
4 | df-ov 7194 | . . . . . . 7 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
5 | 3, 4 | eqtrdi 2787 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | fvconstr.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | fvconst2g 6995 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) | |
9 | 7, 8 | sylan 583 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) |
10 | 6, 9 | eqtrd 2771 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌) |
11 | fvconstr.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
12 | 11 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → 𝑌 ≠ ∅) |
13 | 10, 12 | eqnetrd 2999 | . . 3 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) ≠ ∅) |
14 | 5 | neeq1d 2991 | . . . . 5 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
15 | 14 | biimpa 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) ≠ ∅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅) |
16 | dmxpss 6014 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
17 | ndmfv 6725 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
18 | 17 | necon1ai 2959 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
19 | 16, 18 | sseldi 3885 | . . . 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 | syl5bb 286 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 {csn 4527 〈cop 4533 class class class wbr 5039 × cxp 5534 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 |
This theorem is referenced by: prstchom 45972 prstchom2ALT 45974 |
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