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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfafv23 | Structured version Visualization version GIF version | ||
| Description: A definition of function value in terms of iota, analogous to dffv3 6902. (Contributed by AV, 6-Sep-2022.) |
| Ref | Expression |
|---|---|
| dfafv23 | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfatafv2iota 47222 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
| 2 | dfdfat2 47140 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) | |
| 3 | 2 | simplbi 497 | . . . . . 6 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹) |
| 4 | elimasng 6107 | . . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
| 5 | 3, 4 | sylan 580 | . . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) |
| 6 | df-br 5144 | . . . . 5 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
| 7 | 5, 6 | bitr4di 289 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
| 8 | 7 | elvd 3486 | . . 3 ⊢ (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
| 9 | 8 | iotabidv 6545 | . 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
| 10 | 1, 9 | eqtr4d 2780 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 “ cima 5688 ℩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-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-dfat 47131 df-afv2 47221 |
| This theorem is referenced by: afv2co2 47269 |
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