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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 5076 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | fvconstr.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌})) | |
3 | 2 | oveqd 7301 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵)) |
4 | df-ov 7287 | . . . . . . 7 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) | |
5 | 3, 4 | eqtrdi 2795 | . . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉)) |
7 | fvconstr.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | fvconst2g 7086 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) | |
9 | 7, 8 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = 𝑌) |
10 | 6, 9 | eqtrd 2779 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌) |
11 | fvconstr.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → 𝑌 ≠ ∅) |
13 | 10, 12 | eqnetrd 3012 | . . 3 ⊢ ((𝜑 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴𝐹𝐵) ≠ ∅) |
14 | 5 | neeq1d 3004 | . . . . 5 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅)) |
15 | 14 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) ≠ ∅) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅) |
16 | dmxpss 6079 | . . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅 | |
17 | ndmfv 6813 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) = ∅) | |
18 | 17 | necon1ai 2972 | . . . . 5 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom (𝑅 × {𝑌})) |
19 | 16, 18 | sselid 3920 | . . . 4 ⊢ (((𝑅 × {𝑌})‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑅) |
20 | 15, 19 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) ≠ ∅) → 〈𝐴, 𝐵〉 ∈ 𝑅) |
21 | 13, 20 | impbida 798 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
22 | 1, 21 | syl5bb 283 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2107 ≠ wne 2944 ∅c0 4257 {csn 4562 〈cop 4568 class class class wbr 5075 × cxp 5588 dom cdm 5590 ‘cfv 6437 (class class class)co 7284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-sep 5224 ax-nul 5231 ax-pr 5353 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3435 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-fv 6445 df-ov 7287 |
This theorem is referenced by: prstchom 46369 prstchom2ALT 46371 |
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