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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 7665 | . . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 | |
2 | df-nel 3102 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
3 | eleq1 2893 | . . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) | |
4 | 3 | notbid 310 | . . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
5 | 2, 4 | syl5bb 275 | . . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹)) |
6 | 1, 5 | mpbiri 250 | . 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) |
7 | funressndmafv2rn 42124 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹) | |
8 | 7 | con3i 152 | . . . 4 ⊢ (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴) |
9 | 2, 8 | sylbi 209 | . . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴) |
10 | ndfatafv2 42112 | . . 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 201 | 1 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1658 ∈ wcel 2166 ∉ wnel 3101 𝒫 cpw 4377 ∪ cuni 4657 ran crn 5342 defAt wdfat 42017 ''''cafv2 42109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-iota 6085 df-fun 6124 df-dfat 42020 df-afv2 42110 |
This theorem is referenced by: (None) |
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