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

Theoremonuninsuci 7301* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
𝐴 ∈ On       (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Theoremonsucssi 7302 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵 ↔ suc 𝐴𝐵)

Theoremnlimsucg 7303 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → ¬ Lim suc 𝐴)

Theoremorduninsuc 7304* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
(Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Theoremordunisuc2 7305* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
(Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))

Theoremordzsl 7306* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
(Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))

Theoremonzsl 7307* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))

Theoremdflim3 7308* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))

Theoremdflim4 7309* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))

Theoremlimsuc 7310 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
(Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))

Theoremlimsssuc 7311 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
(Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))

Theoremnlimon 7312* Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)
{𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Theoremlimuni3 7313* The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)

2.4.3  Transfinite induction

Theoremtfi 7314* The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if 𝐴 is a class of ordinal numbers with the property that every ordinal number included in 𝐴 also belongs to 𝐴, then every ordinal number is in 𝐴.

See theorem tfindes 7323 or tfinds 7320 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)

Theoremtfis 7315* Transfinite Induction Schema. If all ordinal numbers less than a given number 𝑥 have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
(𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis2f 7316* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis2 7317* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis3 7318* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝐴 ∈ On → 𝜒)

Theoremtfisi 7319* A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ On)    &   ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)

Theoremtfinds 7320* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ On → (𝜒𝜃))    &   (Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑))       (𝐴 ∈ On → 𝜏)

Theoremtfindsg 7321* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal 𝐵 instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ On → 𝜓)    &   (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))    &   (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))       (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)

Theoremtfindsg2 7322* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal suc 𝐵 instead of zero. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)
(𝑥 = suc 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ On → 𝜓)    &   ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))    &   ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))       ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)

Theoremtfindes 7323* Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
[∅ / 𝑥]𝜑    &   (𝑥 ∈ On → (𝜑[suc 𝑥 / 𝑥]𝜑))    &   (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfinds2 7324* Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff 𝜏 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝜏𝜓)    &   (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))    &   (Lim 𝑥 → (𝜏 → (∀𝑦𝑥 𝜒𝜑)))       (𝑥 ∈ On → (𝜏𝜑))

Theoremtfinds3 7325* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝜂𝜓)    &   (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))    &   (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))       (𝐴 ∈ On → (𝜂𝜏))

2.4.4  The natural numbers (i.e., finite ordinals)

Syntaxcom 7326 Extend class notation to include the class of natural numbers.
class ω

Definitiondf-om 7327* Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e., all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 7328 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 8817, and ω can then be defined per dfom3 8821 (the smallest inductive set) and dfom4 8823.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 5967. They are completely different from the natural numbers (df-nn 11351) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}

Theoremdfom2 7328 An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 7312). (Contributed by NM, 1-Nov-2004.)
ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Theoremelom 7329* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8822. (Contributed by NM, 15-May-1994.)
(𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))

Theoremomsson 7330 Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ω ⊆ On

Theoremlimomss 7331 The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
(Lim 𝐴 → ω ⊆ 𝐴)

Theoremnnon 7332 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theoremnnoni 7333 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
𝐴 ∈ ω       𝐴 ∈ On

Theoremnnord 7334 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
(𝐴 ∈ ω → Ord 𝐴)

Theoremordom 7335 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Ord ω

Theoremelnn 7336 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)

Theoremomon 7337 The class of natural numbers ω is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)
(ω ∈ On ∨ ω = On)

Theoremomelon2 7338 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
(ω ∈ V → ω ∈ On)

Theoremnnlim 7339 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
(𝐴 ∈ ω → ¬ Lim 𝐴)

Theoremomssnlim 7340 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Theoremlimom 7341 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Lim ω

Theorempeano2b 7342 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
(𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)

Theoremnnsuc 7343* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)

Theoremssnlim 7344* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)
((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)

Theoremomsinds 7345* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))       (𝐴 ∈ ω → 𝜒)

2.4.5  Peano's postulates

Theorempeano1 7346 Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7346 through peano5 7350 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.)
∅ ∈ ω

Theorempeano2 7347 The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)

Theorempeano3 7348 The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Theorempeano4 7349 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Theorempeano5 7350* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 7357. (Contributed by NM, 18-Feb-2004.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

Theoremnn0suc 7351* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

2.4.6  Finite induction (for finite ordinals)

Theoremfind 7352* The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that 𝐴 is a set of natural numbers, zero belongs to 𝐴, and given any member of 𝐴 the member's successor also belongs to 𝐴. The conclusion is that every natural number is in 𝐴. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)       𝐴 = ω

Theoremfinds 7353* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝐴 ∈ ω → 𝜏)

Theoremfindsg 7354* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. The basis of this version is an arbitrary natural number 𝐵 instead of zero. (Contributed by NM, 16-Sep-1995.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ ω → 𝜓)    &   (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)

Theoremfinds2 7355* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝜏𝜓)    &   (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))       (𝑥 ∈ ω → (𝜏𝜑))

Theoremfinds1 7356* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝑥 ∈ ω → 𝜑)

Theoremfindes 7357 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. See tfindes 7323 for the transfinite version. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
[∅ / 𝑥]𝜑    &   (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))       (𝑥 ∈ ω → 𝜑)

Theoremdmexg 7358 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → dom 𝐴 ∈ V)

Theoremrnexg 7359 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
(𝐴𝑉 → ran 𝐴 ∈ V)

Theoremdmexd 7360 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)

Theoremdmex 7361 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       dom 𝐴 ∈ V

Theoremrnex 7362 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       ran 𝐴 ∈ V

Theoremiprc 7363 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 21432. (Contributed by NM, 1-Jan-2007.)
¬ I ∈ V

Theoremresiexg 7364 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6735). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(𝐴𝑉 → ( I ↾ 𝐴) ∈ V)

Theoremimaexg 7365 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremimaex 7366 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremexse2 7367 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅𝑉𝑅 Se 𝐴)

Theoremxpexr 7368 If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Theoremxpexr2 7369 If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)
(((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theoremxpexcnv 7370 A condition where the converse of xpex 7223 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)

Theoremsoex 7371 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)
((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)

Theoremelxp4 7372 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7373, elxp6 7462, and elxp7 7463. (Contributed by NM, 17-Feb-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))

Theoremelxp5 7373 Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7372 when the double intersection does not create class existence problems (caused by int0 4711). (Contributed by NM, 1-Aug-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))

Theoremcnvexg 7374 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(𝐴𝑉𝐴 ∈ V)

Theoremcnvex 7375 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
𝐴 ∈ V       𝐴 ∈ V

Theoremrelcnvexb 7376 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))

Theoremf1oexrnex 7377 If the range of a 1-1 onto function is a set, the function itself is a set. (Contributed by AV, 2-Jun-2019.)
((𝐹:𝐴1-1-onto𝐵𝐵𝑉) → 𝐹 ∈ V)

Theoremf1oexbi 7378* There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
(∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)

Theoremcoexg 7379 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoremcoex 7380 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V

Theoremfuncnvuni 7381* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6191 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)

Theoremfun11uni 7382* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝐴 ∧ Fun 𝐴))

Theoremfex2 7383 A function with bounded domain and range is a set. This version of fex 6745 is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)

Theoremfabexg 7384* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)

Theoremfabex 7385* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       𝐹 ∈ V

Theoremdmfex 7386 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)

Theoremf1oabexg 7387* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)

Theoremfun11iun 7388* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)

Theoremffoss 7389* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
𝐹 ∈ V       (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))

Theoremf11o 7390* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
𝐹 ∈ V       (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))

TheoremresfunexgALT 7391 Alternate proof of resfunexg 6735, shorter but requiring ax-pow 5065 and ax-un 7209. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremcofunexg 7392 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremcofunex2g 7393 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)

TheoremfnexALT 7394 Alternate proof of fnex 6737, derived using the Axiom of Replacement in the form of funimaexg 6208. This version uses ax-pow 5065 and ax-un 7209, whereas fnex 6737 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfunrnex 7395 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 6738. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Theoremzfrep6 7396* A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5005 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4994. (Contributed by NM, 10-Oct-2003.)
(∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)

Theoremfornex 7397 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))

Theoremf1dmex 7398 If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4994. (Contributed by NM, 4-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)

Theoremf1ovv 7399 The range of a 1-1 onto function is a set iff its domain is a set. (Contributed by AV, 21-Mar-2019.)
(𝐹:𝐴1-1-onto𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))

Theoremfvclex 7400* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
𝐹 ∈ V       {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V

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