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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2fv0b | Structured version Visualization version GIF version |
Description: The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
afv2fv0b | ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv2fv0 47214 | . 2 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | |
2 | afv20fv0 47212 | . . 3 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) | |
3 | afv2ndeffv0 47209 | . . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅) | |
4 | 2, 3 | jaoi 857 | . 2 ⊢ (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (𝐹‘𝐴) = ∅) |
5 | 1, 4 | impbii 209 | 1 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1536 ∉ wnel 3043 ∅c0 4338 ran crn 5689 ‘cfv 6562 ''''cafv2 47157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 df-dfat 47068 df-afv2 47158 |
This theorem is referenced by: afv2fv0xorb 47216 |
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