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	isorel 7301	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 7302*	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 7303*	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 7304	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 7305	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 7306	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 7307	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 7308	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 7309	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 7310	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 7311	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 7312	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 7313	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 7314	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 7315*	Lemma for isofr 7317. (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 7316*	Lemma for isose 7318. (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 7317	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 7318	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 7319	A weak form of isofr 7317 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 7320	Lemma for isopo 7321. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
Theorem	isopo 7321	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 7322	Lemma for isoso 7323. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	isoso 7323	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 7324	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 7325*	A weak form of isowe 7324 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 7326*	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 7327*	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 7328*	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 7329	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 7330	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7952. (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 7331	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7953. (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 7332*	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 7333	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 7334	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 7335	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 7336	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 7337*	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 7338*	Obsolete version of rexima 7212 as of 14-Aug-2025. Duplicate version of rexima 7212. (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 7339*	Obsolete version of ralima 7211 as of 14-Aug-2025. Duplicate version of ralima 7211. (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 7340	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 7341	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 9094. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7342 for a counterexample. (Use nex 1800 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 7342	Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5270). Specifically, the identity function maps the universe onto its power class. Compare canth 7341 that works for sets. This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3751): 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 4868). See also the remark in ru 3751 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 7343	Extend class notation with restricted description binder.
class (℩𝑥 ∈ 𝐴 𝜑)
Definition	df-riota 7344	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 6464. (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 7345*	Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotabidv 7346*	Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotaeqbidv 7347*	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 7348	Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
⊢ (℩𝑥 ∈ 𝐴 𝜓) ∈ V
Theorem	riotav 7349	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
Theorem	riotauni 7350	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	nfriota1 7351*	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 7352*	Deduction version of nfriota 7356 with a disjoint variable condition, which contrary to nfriotad 7355 does not require ax-13 2370. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	cbvriotaw 7353*	Change bound variable in a restricted description binder. Version of cbvriota 7357 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotavw 7354*	Change bound variable in a restricted description binder. Version of cbvriotav 7358 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 7355	Deduction version of nfriota 7356. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfriotadw 7352 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 7356*	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 7357*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbvriotaw 7353 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 7358*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbvriotavw 7354 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 7359*	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 7360	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 7361*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasbc 7362	Substitution law for descriptions. Compare iotasbc 44408. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
Theorem	riotabidva 7363*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3412 analog.) (Contributed by NM, 17-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotabiia 7364	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3409 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)
Theorem	riota1 7365*	Property of restricted iota. Compare iota1 6488. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))
Theorem	riota1a 7366	Property of iota. (Contributed by NM, 23-Aug-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
Theorem	riota2df 7367*	A deduction version of riota2f 7368. (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 7368*	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 7369*	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 7370*	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 7371*	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 7372*	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 7373*	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 7374*	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 7375*	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))
Theorem	moriotass 7376*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))
Theorem	snriota 7377	A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)})
Theorem	riotaxfrd 7378*	Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5374 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝐶    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)    &   ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)    ⇒   ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)
Theorem	eusvobj2 7379*	Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	eusvobj1 7380*	Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	f1ofveu 7381*	There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)
Theorem	f1ocnvfv3 7382*	Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶))
Theorem	riotaund 7383*	Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)
Theorem	riotassuni 7384*	The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
Theorem	riotaclb 7385*	Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
Theorem	riotarab 7386*	Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒) = (℩𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))
2.3.19  Operations
Syntax	co 7387	Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 11406.)
class (𝐴𝐹𝐵)
Syntax	coprab 7388	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Syntax	cmpo 7389	Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Definition	df-ov 7390	Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. For example, if class 𝐹 is the operation + and arguments 𝐴 and 𝐵 are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 12332). This definition is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets); see ovprc1 7426 and ovprc2 7427. On the other hand, we often find uses for this definition when 𝐹 is a proper class, such as +o in oav 8475. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7391. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
Definition	df-oprab 7391*	Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 7390 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of an operation given by a class abstraction is given by ovmpo 7549. (Contributed by NM, 12-Mar-1995.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Definition	df-mpo 7392*	Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐶(𝑥, 𝑦)". An extension of df-mpt 5189 for two arguments. (Contributed by NM, 17-Feb-2008.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
Theorem	oveq 7393	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Theorem	oveq1 7394	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2 7395	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveq12 7396	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq1i 7397	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
Theorem	oveq2i 7398	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
Theorem	oveq12i 7399	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
Theorem	oveqi 7400	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷)