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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2fv0xorb | Structured version Visualization version GIF version |
Description: If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.) |
Ref | Expression |
---|---|
afv2fv0xorb | ⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv2fv0b 43342 | . 2 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | |
2 | afv2orxorb 43304 | . 2 ⊢ (∅ ∈ ran 𝐹 → (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹))) | |
3 | 1, 2 | syl5bb 284 | 1 ⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∨ wo 841 ⊻ wxo 1495 = wceq 1528 ∈ wcel 2105 ∉ wnel 3120 ∅c0 4288 ran crn 5549 ‘cfv 6348 ''''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-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-xor 1496 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-ne 3014 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-fv 6356 df-dfat 43195 df-afv2 43285 |
This theorem is referenced by: (None) |
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