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Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaint | Structured version Visualization version GIF version |
Description: This is to df-aiota 44192 what iotauni 6333 is to df-iota 6316 (it uses intersection like df-aiota 44192, similar to iotauni 6333 using union like df-iota 6316; we could also prove an analogous result using union here too, in the same way that we have iotaint 6334). (Contributed by BJ, 31-Aug-2024.) |
Ref | Expression |
---|---|
aiotaint | ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuaiotaiota 44195 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
2 | 1 | biimpi 219 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑)) |
3 | iotaint 6334 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
4 | 2, 3 | eqtr3d 2773 | 1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∃!weu 2567 {cab 2714 ∩ cint 4845 ℩cio 6314 ℩'caiota 44190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-pr 4530 df-uni 4806 df-int 4846 df-iota 6316 df-aiota 44192 |
This theorem is referenced by: dfaiota3 44199 |
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