Theorem aiotaint 47886

Description: This is to df-aiota 47880 what iotauni 6517 is to df-iota 6496 (it uses intersection like df-aiota 47880, similar to iotauni 6517 using union like df-iota 6496; we could also prove an analogous result using union here too, in the same way that we have iotaint 6518). (Contributed by BJ, 31-Aug-2024.)

Assertion

Ref	Expression
aiotaint	⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Proof of Theorem aiotaint

Step	Hyp	Ref	Expression
1		reuaiotaiota 47883	. . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
2	1	biimpi 219	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3		iotaint 6518	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
4	2, 3	eqtr3d 2802	1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃!weu 2598  {cab 2743  ∩ cint 4914  ℩cio 6494  ℩'caiota 47878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-sn 4592  df-pr 4594  df-uni 4875  df-int 4915  df-iota 6496  df-aiota 47880

This theorem is used by:  dfaiota3  47887