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Mirrors > Home > ILE Home > Th. List > ndmfvg | GIF version |
Description: The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Ref | Expression |
---|---|
ndmfvg | ⊢ ((A ∈ V ∧ ¬ A ∈ dom 𝐹) → (𝐹‘A) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 1927 | . . . . 5 ⊢ (∃!x A𝐹x → ∃x A𝐹x) | |
2 | eldmg 4473 | . . . . 5 ⊢ (A ∈ V → (A ∈ dom 𝐹 ↔ ∃x A𝐹x)) | |
3 | 1, 2 | syl5ibr 145 | . . . 4 ⊢ (A ∈ V → (∃!x A𝐹x → A ∈ dom 𝐹)) |
4 | 3 | con3d 560 | . . 3 ⊢ (A ∈ V → (¬ A ∈ dom 𝐹 → ¬ ∃!x A𝐹x)) |
5 | tz6.12-2 5112 | . . 3 ⊢ (¬ ∃!x A𝐹x → (𝐹‘A) = ∅) | |
6 | 4, 5 | syl6 29 | . 2 ⊢ (A ∈ V → (¬ A ∈ dom 𝐹 → (𝐹‘A) = ∅)) |
7 | 6 | imp 115 | 1 ⊢ ((A ∈ V ∧ ¬ A ∈ dom 𝐹) → (𝐹‘A) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 Vcvv 2551 ∅c0 3218 class class class wbr 3755 dom cdm 4288 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-dm 4298 df-iota 4810 df-fv 4853 |
This theorem is referenced by: ovprc 5482 |
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