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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmafv2nrn | Structured version Visualization version GIF version | ||
| Description: The value of a class outside its domain is not in the range, compare with ndmfv 6941. (Contributed by AV, 2-Sep-2022.) |
| Ref | Expression |
|---|---|
| ndmafv2nrn | ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | orc 868 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
| 2 | ianor 984 | . . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
| 3 | df-dfat 47131 | . . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 4 | 2, 3 | xchnxbir 333 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) |
| 5 | 1, 4 | sylibr 234 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴) |
| 6 | ndfatafv2nrn 47233 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) | |
| 7 | 5, 6 | syl 17 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∈ wcel 2108 ∉ wnel 3046 {csn 4626 dom cdm 5685 ran crn 5686 ↾ cres 5687 Fun wfun 6555 defAt wdfat 47128 ''''cafv2 47220 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-dfat 47131 df-afv2 47221 |
| This theorem is referenced by: afv2prc 47238 fafv2elrnb 47247 |
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