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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndfatafv2nrn | Structured version Visualization version GIF version |
Description: The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
ndfatafv2nrn | ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndfatafv2 45517 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) | |
2 | pwuninel 8211 | . . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 | |
3 | df-nel 3051 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
4 | eleq1 2826 | . . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) | |
5 | 4 | notbid 318 | . . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
6 | 3, 5 | bitrid 283 | . . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
7 | 2, 6 | mpbiri 258 | . 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
8 | 1, 7 | syl 17 | 1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ∉ wnel 3050 𝒫 cpw 4565 ∪ cuni 4870 ran crn 5639 defAt wdfat 45422 ''''cafv2 45514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nel 3051 df-rab 3411 df-v 3450 df-un 3920 df-in 3922 df-ss 3932 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-uni 4871 df-afv2 45515 |
This theorem is referenced by: ndmafv2nrn 45528 nfunsnafv2 45531 dfatafv2rnb 45533 tz6.12-2-afv2 45543 |
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