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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2ndefb | Structured version Visualization version GIF version |
Description: Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.) |
Ref | Expression |
---|---|
afv2ndefb | ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuninel 7930 | . . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 | |
2 | df-nel 3121 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
3 | eleq1 2897 | . . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) | |
4 | 3 | notbid 319 | . . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
5 | 2, 4 | syl5bb 284 | . . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
6 | 1, 5 | mpbiri 259 | . 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
7 | funressndmafv2rn 43299 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹) | |
8 | 7 | con3i 157 | . . . 4 ⊢ (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴) |
9 | 2, 8 | sylbi 218 | . . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴) |
10 | ndfatafv2 43287 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) |
12 | 6, 11 | impbii 210 | 1 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∉ wnel 3120 𝒫 cpw 4535 ∪ cuni 4830 ran crn 5549 defAt wdfat 43192 ''''cafv2 43284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-dfat 43195 df-afv2 43285 |
This theorem is referenced by: (None) |
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