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Mathbox for Alexander van der Vekens |
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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 6926. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
ndmafv2nrn | ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 865 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
2 | ianor 979 | . . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
3 | df-dfat 46561 | . . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
4 | 2, 3 | xchnxbir 332 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) |
5 | 1, 4 | sylibr 233 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴) |
6 | ndfatafv2nrn 46663 | . 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 394 ∨ wo 845 ∈ wcel 2098 ∉ wnel 3036 {csn 4624 dom cdm 5672 ran crn 5673 ↾ cres 5674 Fun wfun 6536 defAt wdfat 46558 ''''cafv2 46650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nel 3037 df-rab 3420 df-v 3465 df-un 3945 df-in 3947 df-ss 3957 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-uni 4904 df-dfat 46561 df-afv2 46651 |
This theorem is referenced by: afv2prc 46668 fafv2elrnb 46677 |
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