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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovg | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain, analogous to ndmovg 7141. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaovg | ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5436 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
2 | eleq2 2848 | . . . . . 6 ⊢ ((𝑅 × 𝑆) = dom 𝐹 → (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) | |
3 | 2 | eqcoms 2780 | . . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
4 | 1, 3 | syl5bbr 277 | . . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
5 | 4 | notbid 310 | . . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
6 | 5 | biimpa 469 | . 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
7 | ndmaov 42734 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
8 | 6, 7 | syl 17 | 1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 〈cop 4441 × cxp 5398 dom cdm 5400 ((caov 42669 |
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 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 |
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 2014 df-mo 2544 df-eu 2580 df-clab 2754 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 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-int 4744 df-br 4924 df-opab 4986 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-res 5412 df-iota 6146 df-fun 6184 df-fv 6190 df-aiota 42637 df-dfat 42670 df-afv 42671 df-aov 42672 |
This theorem is referenced by: (None) |
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