|   | Mathbox for Zhi Wang | < Previous  
      Next > Nearby theorems | |
| 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 4342 | . . 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 2793 | . . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) | 
| 7 | 6 | neeq1d 3000 | . . . 4 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) | 
| 8 | dmxpss 6191 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
| 9 | ndmfv 6941 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
| 10 | 9 | necon1ai 2968 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) | 
| 11 | 8, 10 | sselid 3981 | . . . 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 5144 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 15 | 13, 14 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴𝑅𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 dom cdm 5685 ‘cfv 6561 (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 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: prsthinc 49111 | 
| Copyright terms: Public domain | W3C validator |