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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2ex | Structured version Visualization version GIF version |
Description: The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2ex | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfatafv2iota 47159 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
2 | iotaex 6535 | . 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 1, 2 | eqeltrdi 2846 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3477 class class class wbr 5147 ℩cio 6513 defAt wdfat 47065 ''''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-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-uni 4912 df-iota 6515 df-afv2 47158 |
This theorem is referenced by: dfatbrafv2b 47194 fnbrafv2b 47197 dfatdmfcoafv2 47203 dfatcolem 47204 |
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