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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovcom | Structured version Visualization version GIF version |
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7149. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
Ref | Expression |
---|---|
ndmaovcom | ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5439 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
2 | ndmaov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
3 | 2 | eqcomi 2781 | . . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹 |
4 | 3 | eleq2i 2851 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
5 | 1, 4 | bitr3i 269 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmaov 42813 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
7 | 5, 6 | sylnbi 322 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = V) |
8 | ancom 453 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
9 | opelxp 5439 | . . . 4 ⊢ (〈𝐵, 𝐴〉 ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
10 | 3 | eleq2i 2851 | . . . 4 ⊢ (〈𝐵, 𝐴〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐵, 𝐴〉 ∈ dom 𝐹) |
11 | 8, 9, 10 | 3bitr2i 291 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐵, 𝐴〉 ∈ dom 𝐹) |
12 | ndmaov 42813 | . . 3 ⊢ (¬ 〈𝐵, 𝐴〉 ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V) | |
13 | 11, 12 | sylnbi 322 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐵𝐹𝐴)) = V) |
14 | 7, 13 | eqtr4d 2811 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3409 〈cop 4441 × cxp 5401 dom cdm 5403 ((caov 42748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-int 4746 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-res 5415 df-iota 6149 df-fun 6187 df-fv 6193 df-aiota 42716 df-dfat 42749 df-afv 42750 df-aov 42751 |
This theorem is referenced by: (None) |
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