Theorem sn-iotalemcor 42709

Description: Corollary of sn-iotalem 42708. Compare sb8iota 6452. (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 42708	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
2	1	unieqi 4850	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
3		df-iota 6441	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4		df-iota 6441	. 2 ⊢ (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
5	2, 3, 4	3eqtr4i 2772	1 ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   = wceq 1547  {cab 2717  {csn 4555  ∪ cuni 4838  ℩cio 6439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-sn 4556  df-uni 4839  df-iota 6441

This theorem is referenced by: (None)