Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 6725. (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 44226 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | 1 | simplbi 501 | . 2 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹) |
3 | afvnfundmuv 44246 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V) | |
4 | 2, 3 | nsyl5 162 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 dom cdm 5536 ↾ cres 5538 Fun wfun 6352 defAt wdfat 44223 '''cafv 44224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-iota 6316 df-fun 6360 df-fv 6366 df-aiota 44192 df-dfat 44226 df-afv 44227 |
This theorem is referenced by: afvvdm 44248 afvprc 44251 afvco2 44283 ndmaov 44290 aovprc 44295 |
Copyright terms: Public domain | W3C validator |