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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 45571 | . 2 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | |
2 | afv20fv0 45569 | . . 3 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) | |
3 | afv2ndeffv0 45566 | . . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅) | |
4 | 2, 3 | jaoi 856 | . 2 ⊢ (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (𝐹‘𝐴) = ∅) |
5 | 1, 4 | impbii 208 | 1 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∉ wnel 3050 ∅c0 4287 ran crn 5639 ‘cfv 6501 ''''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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fv 6509 df-dfat 45425 df-afv2 45515 |
This theorem is referenced by: afv2fv0xorb 45573 |
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