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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 47222 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
| 2 | iotaex 6534 | . 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
| 3 | 1, 2 | eqeltrdi 2849 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ℩cio 6512 defAt wdfat 47128 ''''cafv2 47220 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-afv2 47221 |
| This theorem is referenced by: dfatbrafv2b 47257 fnbrafv2b 47260 dfatdmfcoafv2 47266 dfatcolem 47267 |
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