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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvconstr2 | 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})) |
fvconstr2.2 | ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵)) |
Ref | Expression |
---|---|
fvconstr2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvconstr2.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵)) | |
2 | 1 | ne0d 4348 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ≠ ∅) |
3 | fvconstr.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
4 | 3 | oveqd 7448 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
5 | df-ov 7434 | . . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
6 | 4, 5 | eqtrdi 2791 | . . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | 6 | neeq1d 2998 | . . . 4 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
8 | dmxpss 6193 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
9 | ndmfv 6942 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
10 | 9 | necon1ai 2966 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
11 | 8, 10 | sselid 3993 | . . . 4 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅) |
12 | 7, 11 | biimtrdi 253 | . . 3 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅)) |
13 | 2, 12 | mpd 15 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑅) |
14 | df-br 5149 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
15 | 13, 14 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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-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-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: prsthinc 48855 |
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