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	tfisi 7801*	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 7802*	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 7803*	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 7804*	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 7805*	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 7806*	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 7807*	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 7808	Extend class notation to include the class of natural numbers.
class ω
Definition	df-om 7809*	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 7810 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9553, and ω can then be defined per dfom3 9557 (the smallest inductive set) and dfom4 9559. Note: the natural numbers ω are a subset of the ordinal numbers df-on 6319. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12164) 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 7810	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 7793). (Contributed by NM, 1-Nov-2004.)
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Theorem	elom 7811*	Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9558. (Contributed by NM, 15-May-1994.)
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
Theorem	omsson 7812	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ω ⊆ On
Theorem	limomss 7813	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 7814	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 7815	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 7816	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	trom 7817	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 7818	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 7819	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	omon 7820	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 7821	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	nnlim 7822	A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
⊢ (𝐴 ∈ ω → ¬ Lim 𝐴)
Theorem	omssnlim 7823	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 7824	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 7825	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 7826*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	omsucne 7827	A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
Theorem	ssnlim 7828*	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 7829*	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 7830	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 7831	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 7831 through peano5 7835 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 7680. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ∅ ∈ ω
Theorem	peano2 7832	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 7833	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 7834	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 7835*	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 7842. (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 7836*	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 7837*	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 7838*	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 7839*	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 7840*	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 7841*	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 7842	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 7805 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 7843	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 7844	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 7845	The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 ∈ V)
Theorem	fndmexd 7846	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 7847	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 7848	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 7849	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 7850	Alternate proof of dmfex 7847: shorter but using ax-rep 5212. (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 7851	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 7852	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 7853	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 23231. (Contributed by NM, 1-Jan-2007.)
⊢ ¬ I ∈ V
Theorem	resiexg 7854	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7161). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
Theorem	imaexg 7855	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 7856	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 7857	The range of a set is a set. Deduction version of rnexd 7857. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran 𝐴 ∈ V)
Theorem	imaexd 7858	The image of a set is a set. Deduction version of imaexg 7855. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)
Theorem	exse2 7859	Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	xpexr 7860	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 7861	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 7862	A condition where the converse of xpex 7698 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 7863	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 7864	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7865, elxp6 7967, and elxp7 7968. (Contributed by NM, 17-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	elxp5 7865	Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7864 when the double intersection does not create class existence problems (caused by int0 4905). (Contributed by NM, 1-Aug-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	cnvexg 7866	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 7867	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 7868	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 7869	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 7870*	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 7871	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 7872	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V
Theorem	coexd 7873	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 7874*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6559 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)
Theorem	fun11uni 7875*	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 7876	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 7877	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 7878	A function with bounded domain and codomain is a set. This version of fex 7172 is proven without the Axiom of Replacement ax-rep 5212, but depends on ax-un 7680, which is not required for the proof of fex 7172. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
Theorem	fabexd 7879*	Existence of a set of functions. In contrast to fabex 7882 or fabexg 7880, 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 7880*	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 7881*	Obsolete version of fabexg 7880 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 7882*	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 7883*	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 7884*	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 7885*	Obsolete version of f1oabexg 7884 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 7886*	Lemma for fiun 7887 and f1iun 7888. Formerly part of f1iun 7888. (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 7887*	The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7888. (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 7888*	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 7889*	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 7890*	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))
Theorem	f11o 7891*	Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
Theorem	resfunexgALT 7892	Alternate proof of resfunexg 7161, shorter but requiring ax-pow 5300 and ax-un 7680. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)
Theorem	cofunexg 7893	Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	cofunex2g 7894	Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	fnexALT 7895	Alternate proof of fnex 7163, derived using the Axiom of Replacement in the form of funimaexg 6577. This version uses ax-pow 5300 and ax-un 7680, whereas fnex 7163 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)
Theorem	funexw 7896	Weak version of funex 7165 that holds without ax-rep 5212. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V)
Theorem	mptexw 7897*	Weak version of mptex 7169 that holds without ax-rep 5212. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
Theorem	funrnex 7898	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 7165. (Contributed by NM, 11-Nov-1995.)
⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Theorem	zfrep6OLD 7899*	Obsolete proof of zfrep6 5224 as of 5-Apr-2026. (Contributed by NM, 10-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	focdmex 7900	If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))