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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvvdm | Structured version Visualization version GIF version |
Description: If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvvdm | ⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmafv 43330 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) | |
2 | nvelim 43313 | . . 3 ⊢ ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹'''𝐴) ∈ 𝐵) |
4 | 3 | con4i 114 | 1 ⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1531 ∈ wcel 2108 Vcvv 3493 dom cdm 5548 '''cafv 43307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-aiota 43276 df-dfat 43309 df-afv 43310 |
This theorem is referenced by: aovvdm 43375 aovrcl 43379 aoprssdm 43392 |
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