P. 79 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	limsuc 7801	The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
Theorem	limsssuc 7802	A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))
Theorem	nlimon 7803*	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}
Theorem	limuni3 7804*	The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)
2.4.3  Transfinite induction
Theorem	tfi 7805*	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 7815 or tfinds 7812 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)
Theorem	tfisg 7806*	A closed form of tfis 7807. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)
Theorem	tfis 7807*	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 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2f 7808*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2 7809*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis3 7810*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜒)
Theorem	tfisi 7811*	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ On)    &   ⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	tfinds 7812*	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. Theorem 1.19 of [Schloeder] p. 3. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ On → (𝜒 → 𝜃))    &   ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜏)
Theorem	tfindsg 7813*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))    &   ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → 𝜏)
Theorem	tfindsg2 7814*	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. (Contributed by NM, 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = suc 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))    &   ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)
Theorem	tfindes 7815*	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 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfinds2 7816*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜏 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜏 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))
Theorem	tfinds3 7817*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝜂 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))
2.4.4  The natural numbers (i.e., finite ordinals)
Syntax	com 7818	Extend class notation to include the class of natural numbers.
class ω
Definition	df-om 7819*	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 7820 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9564, and ω can then be defined per dfom3 9568 (the smallest inductive set) and dfom4 9570. Note: the natural numbers ω are a subset of the ordinal numbers df-on 6329. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12158) with analogous properties and operations, but they will be different sets. (Contributed by NM, 15-May-1994.)
⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom2 7820	An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the restricted class abstraction of non-limit ordinal numbers (see nlimon 7803). (Contributed by NM, 1-Nov-2004.)
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Theorem	elom 7821*	Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9569. (Contributed by NM, 15-May-1994.)
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
Theorem	omsson 7822	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ω ⊆ On
Theorem	limomss 7823	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)
Theorem	nnon 7824	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 7825	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 7826	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	trom 7827	The class of finite ordinals ω is a transitive class. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Tr ω
Theorem	ordom 7828	The class of finite ordinals ω is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Ord ω
Theorem	elnn 7829	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	omon 7830	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)
Theorem	omelon2 7831	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	nnlim 7832	A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
⊢ (𝐴 ∈ ω → ¬ Lim 𝐴)
Theorem	omssnlim 7833	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}
Theorem	limom 7834	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Theorem 1.23 of [Schloeder] p. 4. 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
Theorem	peano2b 7835	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 7836*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	omsucne 7837	A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
Theorem	ssnlim 7838*	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 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
Theorem	omsinds 7839*	Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ ω → 𝜒)
Theorem	omun 7840	The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)
2.4.5  Peano's postulates
Theorem	peano1 7841	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 7841 through peano5 7845 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.) Avoid ax-un 7690. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ∅ ∈ ω
Theorem	peano2 7842	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)
Theorem	peano3 7843	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 ≠ ∅)
Theorem	peano4 7844	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 𝐵 ↔ 𝐴 = 𝐵))
Theorem	peano5 7845*	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 7852. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2147, ax-12 2185. (Revised by GG, 3-Oct-2024.)
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	nn0suc 7846*	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)
Theorem	find 7847*	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.) (Proof shortened by Wolf Lammen, 28-May-2024.)
⊢ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)    ⇒   ⊢ 𝐴 = ω
Theorem	finds 7848*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ω → 𝜏)
Theorem	findsg 7849*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ ω → 𝜓)    &   ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))    ⇒   ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → 𝜏)
Theorem	finds2 7850*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))
Theorem	finds1 7851*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
Theorem	findes 7852	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 7815 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)
2.4.7  Relations and functions (cont.)
Theorem	dmexg 7853	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
Theorem	rnexg 7854	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.)
⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
Theorem	dmexd 7855	The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 ∈ V)
Theorem	fndmexd 7856	If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	dmfex 7857	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.)
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
Theorem	fndmexb 7858	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
Theorem	fdmexb 7859	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
Theorem	dmfexALT 7860	Alternate proof of dmfex 7857: shorter but using ax-rep 5226. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 23-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
Theorem	dmex 7861	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
Theorem	rnex 7862	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
Theorem	iprc 7863	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 23213. (Contributed by NM, 1-Jan-2007.)
⊢ ¬ I ∈ V
Theorem	resiexg 7864	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7171). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
Theorem	imaexg 7865	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
Theorem	imaex 7866	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 “ 𝐵) ∈ V
Theorem	rnexd 7867	The range of a set is a set. Deduction version of rnexd 7867. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran 𝐴 ∈ V)
Theorem	imaexd 7868	The image of a set is a set. Deduction version of imaexg 7865. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)
Theorem	exse2 7869	Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	xpexr 7870	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))
Theorem	xpexr2 7871	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))
Theorem	xpexcnv 7872	A condition where the converse of xpex 7708 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
Theorem	soex 7873	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 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)
Theorem	elxp4 7874	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7875, elxp6 7977, and elxp7 7978. (Contributed by NM, 17-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	elxp5 7875	Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7874 when the double intersection does not create class existence problems (caused by int0 4919). (Contributed by NM, 1-Aug-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	cnvexg 7876	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
Theorem	cnvex 7877	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
Theorem	relcnvexb 7878	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))
Theorem	f1oexrnex 7879	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:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ V)
Theorem	f1oexbi 7880*	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:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
Theorem	coexg 7881	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 7882	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V
Theorem	coexd 7883	The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)
Theorem	funcnvuni 7884*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6569 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)
Theorem	fun11uni 7885*	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.)
⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ∪ 𝐴 ∧ Fun ◡∪ 𝐴))
Theorem	resf1extb 7886	Extension of an injection which is a restriction of a function. (Contributed by AV, 3-Oct-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵))
Theorem	resf1ext2b 7887	Extension of an injection which is a restriction of a function. (Contributed by AV, 3-Oct-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
Theorem	fex2 7888	A function with bounded domain and codomain is a set. This version of fex 7182 is proven without the Axiom of Replacement ax-rep 5226, but depends on ax-un 7690, which is not required for the proof of fex 7182. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
Theorem	fabexd 7889*	Existence of a set of functions. In contrast to fabex 7892 or fabexg 7890, the condition in the class abstraction does not contain the function explicitly, but the function can be derived from it. Therefore, this theorem is also applicable for more special functions like one-to-one, onto or one-to-one onto functions. (Contributed by AV, 20-May-2025.)
⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)
Theorem	fabexg 7890*	Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)
⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	fabexgOLD 7891*	Obsolete version of fabexg 7890 as of 9-Jun-2025. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	fabex 7892*	Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ⊢ 𝐹 ∈ V
Theorem	mapex 7893*	The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) (Proof shortened by AV, 16-Jun-2025.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
Theorem	f1oabexg 7894*	The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	f1oabexgOLD 7895*	Obsolete version of f1oabexg 7894 as of 9-Jun-2025. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	fiunlem 7896*	Lemma for fiun 7897 and f1iun 7898. Formerly part of f1iun 7898. (Contributed by AV, 6-Oct-2023.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
Theorem	fiun 7897*	The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7898. (Contributed by AV, 6-Oct-2023.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷⟶𝑆)
Theorem	f1iun 7898*	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.) (Proof shortened by AV, 5-Nov-2023.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷–1-1→𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷–1-1→𝑆)
Theorem	fviunfun 7899*	The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.)
⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)    ⇒   ⊢ ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))
Theorem	ffoss 7900*	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))