Theorem sn-iotalemcor 42263

Description: Corollary of sn-iotalem 42262. Compare sb8iota 6448. (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 42262	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
2	1	unieqi 4868	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
3		df-iota 6437	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4		df-iota 6437	. 2 ⊢ (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
5	2, 3, 4	3eqtr4i 2764	1 ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   = wceq 1541  {cab 2709  {csn 4573  ∪ cuni 4856  ℩cio 6435

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3914  df-sn 4574  df-uni 4857  df-iota 6437

This theorem is referenced by: (None)