Theorem aiotaint 47646

Description: This is to df-aiota 47640 what iotauni 6493 is to df-iota 6472 (it uses intersection like df-aiota 47640, similar to iotauni 6493 using union like df-iota 6472; we could also prove an analogous result using union here too, in the same way that we have iotaint 6494). (Contributed by BJ, 31-Aug-2024.)

Assertion

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

Proof of Theorem aiotaint

Step	Hyp	Ref	Expression
1		reuaiotaiota 47643	. . 3 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
2	1	biimpi 218	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3		iotaint 6494	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
4	2, 3	eqtr3d 2798	1 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1559  ∃!weu 2594  {cab 2739  ∩ cint 4902  ℩cio 6470  ℩'caiota 47638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-sn 4580  df-pr 4582  df-uni 4863  df-int 4903  df-iota 6472  df-aiota 47640

This theorem is referenced by:  dfaiota3  47647