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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovrcl | Structured version Visualization version GIF version |
Description: Reverse closure law, in contrast to ndmovrcl 7350 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
Ref | Expression |
---|---|
ndmaovrcl | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aovvdm 44210 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
2 | opelxp 5561 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
3 | 2 | biimpi 219 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
4 | ndmaov.1 | . . 3 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | 3, 4 | eleq2s 2851 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
6 | 1, 5 | syl 17 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 〈cop 4522 × cxp 5523 dom cdm 5525 ((caov 44143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-int 4837 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-res 5537 df-iota 6297 df-fun 6341 df-fv 6347 df-aiota 44109 df-dfat 44144 df-afv 44145 df-aov 44146 |
This theorem is referenced by: (None) |
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