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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmafv | Structured version Visualization version GIF version |
Description: The value of a class outside its domain is the universe, compare with ndmfv 6936. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
ndmafv | ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dfat 47019 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | 1 | simplbi 497 | . 2 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹) |
3 | afvnfundmuv 47039 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V) | |
4 | 2, 3 | nsyl5 159 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1535 ∈ wcel 2104 Vcvv 3477 {csn 4630 dom cdm 5683 ↾ cres 5685 Fun wfun 6552 defAt wdfat 47016 '''cafv 47017 |
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 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-br 5150 df-opab 5212 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-res 5695 df-iota 6510 df-fun 6560 df-fv 6566 df-aiota 46985 df-dfat 47019 df-afv 47020 |
This theorem is referenced by: afvvdm 47041 afvprc 47044 afvco2 47076 ndmaov 47083 aovprc 47088 |
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