Theorem aiotaint 47079

Description: This is to df-aiota 47073 what iotauni 6463 is to df-iota 6442 (it uses intersection like df-aiota 47073, similar to iotauni 6463 using union like df-iota 6442; we could also prove an analogous result using union here too, in the same way that we have iotaint 6464). (Contributed by BJ, 31-Aug-2024.)

Assertion

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

Proof of Theorem aiotaint

Step	Hyp	Ref	Expression
1		reuaiotaiota 47076	. . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
2	1	biimpi 216	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3		iotaint 6464	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
4	2, 3	eqtr3d 2766	1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1540  ∃!weu 2561  {cab 2707  ∩ cint 4899  ℩cio 6440  ℩'caiota 47071

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-sn 4580  df-pr 4582  df-uni 4862  df-int 4900  df-iota 6442  df-aiota 47073

This theorem is referenced by:  dfaiota3  47080