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| Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaint | Structured version Visualization version GIF version | ||
| Description: This is to df-aiota 47097 what iotauni 6536 is to df-iota 6514 (it uses intersection like df-aiota 47097, similar to iotauni 6536 using union like df-iota 6514; we could also prove an analogous result using union here too, in the same way that we have iotaint 6537). (Contributed by BJ, 31-Aug-2024.) |
| Ref | Expression |
|---|---|
| aiotaint | ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reuaiotaiota 47100 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑)) |
| 3 | iotaint 6537 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
| 4 | 2, 3 | eqtr3d 2779 | 1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∃!weu 2568 {cab 2714 ∩ cint 4946 ℩cio 6512 ℩'caiota 47095 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 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-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-iota 6514 df-aiota 47097 |
| This theorem is referenced by: dfaiota3 47104 |
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