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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 5041 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | fvconstr.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
3 | 2 | oveqd 7199 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
4 | df-ov 7185 | . . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
5 | 3, 4 | eqtrdi 2790 | . . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | fvconstr.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | fvconst2g 6986 | . . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) | |
9 | 7, 8 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) |
10 | 6, 9 | eqtrd 2774 | . . 3 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌) |
11 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌) | |
12 | fvconstr.3 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅) |
14 | 11, 13 | eqnetrd 3002 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅) |
15 | 5 | neeq1d 2994 | . . . . . 6 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
16 | 15 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
17 | 14, 16 | mpbid 235 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅) |
18 | dmxpss 6013 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
19 | ndmfv 6716 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
20 | 19 | necon1ai 2962 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
21 | 18, 20 | sseldi 3885 | . . . 4 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅) |
22 | 17, 21 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 〈𝐴, 𝐵〉 ∈ 𝑅) |
23 | 10, 22 | impbida 801 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌)) |
24 | 1, 23 | syl5bb 286 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∅c0 4221 {csn 4526 〈cop 4532 class class class wbr 5040 × cxp 5533 dom cdm 5535 ‘cfv 6349 (class class class)co 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7185 |
This theorem is referenced by: prsthinc 45846 prstchom2ALT 45864 |
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