Theorem aiotaint 47337

Description: This is to df-aiota 47331 what iotauni 6469 is to df-iota 6448 (it uses intersection like df-aiota 47331, similar to iotauni 6469 using union like df-iota 6448; we could also prove an analogous result using union here too, in the same way that we have iotaint 6470). (Contributed by BJ, 31-Aug-2024.)

Assertion

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

Proof of Theorem aiotaint

Step	Hyp	Ref	Expression
1		reuaiotaiota 47334	. . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
2	1	biimpi 216	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3		iotaint 6470	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
4	2, 3	eqtr3d 2773	1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1541  ∃!weu 2568  {cab 2714  ∩ cint 4902  ℩cio 6446  ℩'caiota 47329

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 2184  ax-ext 2708

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-sn 4581  df-pr 4583  df-uni 4864  df-int 4903  df-iota 6448  df-aiota 47331

This theorem is referenced by:  dfaiota3  47338