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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 2067 | . . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥) | |
2 | eldmg 4836 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥)) | |
3 | 1, 2 | imbitrrid 156 | . . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹)) |
4 | 3 | con3d 632 | . . 3 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥)) |
5 | tz6.12-2 5520 | . . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) | |
6 | 4, 5 | syl6 33 | . 2 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) |
7 | 6 | imp 124 | 1 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1363 ∃wex 1502 ∃!weu 2037 ∈ wcel 2159 Vcvv 2751 ∅c0 3436 class class class wbr 4017 dom cdm 4640 ‘cfv 5230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-v 2753 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-dm 4650 df-iota 5192 df-fv 5238 |
This theorem is referenced by: ovprc 5925 sumnul 11449 |
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