P. 74 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	f1eqcocnv 7301	Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Proof shortened by Wolf Lammen, 29-May-2024.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
Theorem	fveqf1o 7302	Given a bijection 𝐹, produce another bijection 𝐺 which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))    ⇒   ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺:𝐴–1-1-onto→𝐵 ∧ (𝐺‘𝐶) = 𝐷))
Theorem	f1ocoima 7303	The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))
Theorem	nf1const 7304	A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.)
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶)
Theorem	nf1oconst 7305	A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.)
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1-onto→𝐶)
Theorem	f1ofvswap 7306	Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵)
Theorem	fvf1pr 7307	Values of a one-to-one function between two sets with two elements. Actually, such a function is a bijection. (Contributed by AV, 22-Jul-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))
Theorem	fliftrel 7308*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
Theorem	fliftel 7309*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
Theorem	fliftel1 7310*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵)
Theorem	fliftcnv 7311*	Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
Theorem	fliftfun 7312*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷)))
Theorem	fliftfund 7313*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 = 𝐶)) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fliftfuns 7314*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
Theorem	fliftf 7315*	The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))
Theorem	fliftval 7316*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)
Theorem	isoeq1 7317	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq2 7318	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq3 7319	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))
Theorem	isoeq4 7320	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
Theorem	isoeq5 7321	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))
Theorem	nfiso 7322	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎ𝑥𝐻    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Theorem	isof1o 7323	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
Theorem	isof1oidb 7324	A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)
⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))
Theorem	isof1oopb 7325	A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.)
⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
Theorem	isorel 7326	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	soisores 7327*	Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))
Theorem	soisoi 7328*	Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	isoid 7329	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Theorem	isocnv 7330	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
Theorem	isocnv2 7331	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
Theorem	isocnv3 7332	Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)    &   ⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)    ⇒   ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))
Theorem	isores2 7333	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
Theorem	isores1 7334	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
Theorem	isores3 7335	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Theorem	isotr 7336	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))
Theorem	isomin 7337	Isomorphisms preserve minimal elements. Note that (◡𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥 ∣ 𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
Theorem	isoini 7338	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
Theorem	isoini2 7339	Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))    &   ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))    ⇒   ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Theorem	isofrlem 7340*	Lemma for isofr 7342. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	isoselem 7341*	Lemma for isose 7343. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
Theorem	isofr 7342	An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
Theorem	isose 7343	An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))
Theorem	isofr2 7344	A weak form of isofr 7342 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	isopolem 7345	Lemma for isopo 7346. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
Theorem	isopo 7346	An isomorphism preserves the property of being a partial order. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))
Theorem	isosolem 7347	Lemma for isoso 7348. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	isoso 7348	An isomorphism preserves the property of being a strict total order. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
Theorem	isowe 7349	An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
Theorem	isowe2 7350*	A weak form of isowe 7349 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	f1oiso 7351*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	f1oiso2 7352*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))}    ⇒   ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	f1owe 7353*	Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	weniso 7354	A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 Isom 𝑅, 𝑅 (𝐴, 𝐴)) → 𝐹 = ( I ↾ 𝐴))
Theorem	weisoeq 7355	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7971. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	weisoeq2 7356	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7972. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	knatar 7357*	The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)
⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))
Theorem	fvresval 7358	The value of a restricted function at a class is either the empty set or the value of the unrestricted function at that class. (Contributed by Scott Fenton, 4-Sep-2011.)
⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
Theorem	funeldmb 7359	If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))
Theorem	eqfunresadj 7360	Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))
Theorem	eqfunressuc 7361	Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))
Theorem	fnssintima 7362*	Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
Theorem	imaeqsexvOLD 7363*	Obsolete version of rexima 7238 as of 14-Aug-2025. Duplicate version of rexima 7238. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	imaeqsalvOLD 7364*	Obsolete version of ralima 7237 as of 14-Aug-2025. Duplicate version of ralima 7237. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
Theorem	fnimasnd 7365	The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})
2.3.17  Cantor's Theorem
Theorem	canth 7366	No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 9119. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7367 for a counterexample. (Use nex 1830 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴
Theorem	ncanth 7367	Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5284). Specifically, the identity function maps the universe onto its power class. Compare canth 7366 that works for sets. This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3744): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4870). See also the remark in ru 3744 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)
⊢ I :V–onto→𝒫 V
2.3.18  Restricted iota (description binder)
Syntax	crio 7368	Extend class notation with restricted description binder.
class (℩𝑥 ∈ 𝐴 𝜑)
Definition	df-riota 7369	Define restricted description binder. In case there is no unique 𝑥 such that (𝑥 ∈ 𝐴 ∧ 𝜑) holds, it evaluates to the empty set. See also comments for df-iota 6494. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	riotaeqdv 7370*	Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotabidv 7371*	Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotaeqbidv 7372*	Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))
Theorem	riotaex 7373	Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
⊢ (℩𝑥 ∈ 𝐴 𝜓) ∈ V
Theorem	riotav 7374	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
Theorem	riotauni 7375	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	nfriota1 7376*	The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
Theorem	nfriotadw 7377*	Deduction version of nfriota 7381 with a disjoint variable condition, which contrary to nfriotad 7380 does not require ax-13 2404. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	cbvriotaw 7378*	Change bound variable in a restricted description binder. Version of cbvriota 7382 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotavw 7379*	Change bound variable in a restricted description binder. Version of cbvriotav 7383 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	nfriotad 7380	Deduction version of nfriota 7381. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker nfriotadw 7377 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	nfriota 7381*	A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)
Theorem	cbvriota 7382*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbvriotaw 7378 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotav 7383*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbvriotavw 7379 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	csbriota 7384*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
Theorem	riotacl2 7385	Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	riotacl 7386*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasbc 7387	Substitution law for descriptions. Compare iotasbc 45112. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
Theorem	riotabidva 7388*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3422 analog.) (Contributed by NM, 17-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotabiia 7389	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3420 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)
Theorem	riota1 7390*	Property of restricted iota. Compare iota1 6517. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))
Theorem	riota1a 7391	Property of iota. (Contributed by NM, 23-Aug-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
Theorem	riota2df 7392*	A deduction version of riota2f 7393. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
Theorem	riota2f 7393*	This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
Theorem	riota2 7394*	This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
Theorem	riotaeqimp 7395*	If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)    &   ⊢ 𝐽 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)    &   ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)    &   ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)
Theorem	riotaprop 7396*	Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑)    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))
Theorem	riota5f 7397*	A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)
Theorem	riota5 7398*	A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)
Theorem	riotass2 7399*	Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotass 7400*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))