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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovvdm | Structured version Visualization version GIF version |
Description: If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovvdm | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 47071 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2830 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvdm 47091 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
4 | 2, 3 | sylbi 217 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 〈cop 4637 dom cdm 5689 '''cafv 47067 ((caov 47068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-aiota 47035 df-dfat 47069 df-afv 47070 df-aov 47071 |
This theorem is referenced by: ndmaovrcl 47154 ndmaovass 47156 ndmaovdistr 47157 |
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