Theorem sn-iotalemcor 42841

Description: Corollary of sn-iotalem 42840. Compare sb8iota 6488. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
sn-iotalemcor	⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotalemcor

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sn-iotalem 42840	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
2	1	unieqi 4877	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
3		df-iota 6477	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4		df-iota 6477	. 2 ⊢ (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
5	2, 3, 4	3eqtr4i 2795	1 ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   = wceq 1560  {cab 2740  {csn 4582  ∪ cuni 4865  ℩cio 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-sn 4583  df-uni 4866  df-iota 6477

This theorem is referenced by: (None)