![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaint | Structured version Visualization version GIF version |
Description: This is to df-aiota 47034 what iotauni 6537 is to df-iota 6515 (it uses intersection like df-aiota 47034, similar to iotauni 6537 using union like df-iota 6515; we could also prove an analogous result using union here too, in the same way that we have iotaint 6538). (Contributed by BJ, 31-Aug-2024.) |
Ref | Expression |
---|---|
aiotaint | ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuaiotaiota 47037 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
2 | 1 | biimpi 216 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑)) |
3 | iotaint 6538 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
4 | 2, 3 | eqtr3d 2776 | 1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∃!weu 2565 {cab 2711 ∩ cint 4950 ℩cio 6513 ℩'caiota 47032 |
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-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-sn 4631 df-pr 4633 df-uni 4912 df-int 4951 df-iota 6515 df-aiota 47034 |
This theorem is referenced by: dfaiota3 47041 |
Copyright terms: Public domain | W3C validator |