Theorem aiotaint 47253

Description: This is to df-aiota 47247 what iotauni 6466 is to df-iota 6445 (it uses intersection like df-aiota 47247, similar to iotauni 6466 using union like df-iota 6445; we could also prove an analogous result using union here too, in the same way that we have iotaint 6467). (Contributed by BJ, 31-Aug-2024.)

Assertion

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

Proof of Theorem aiotaint

Step	Hyp	Ref	Expression
1		reuaiotaiota 47250	. . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
2	1	biimpi 216	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3		iotaint 6467	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
4	2, 3	eqtr3d 2770	1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1541  ∃!weu 2565  {cab 2711  ∩ cint 4899  ℩cio 6443  ℩'caiota 47245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-sn 4578  df-pr 4580  df-uni 4861  df-int 4900  df-iota 6445  df-aiota 47247

This theorem is referenced by:  dfaiota3  47254