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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 7334 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 43404 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
2 | opelxp 5591 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
3 | 2 | biimpi 218 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
4 | ndmaov.1 | . . 3 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | 3, 4 | eleq2s 2931 | . 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 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 dom cdm 5555 ((caov 43337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-aiota 43305 df-dfat 43338 df-afv 43339 df-aov 43340 |
This theorem is referenced by: (None) |
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