P. 80 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	findes 7901	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 7863 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 7902	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 7903	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 7904	The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 ∈ V)
Theorem	fndmexd 7905	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 7906	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 7907	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 7908	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 7909	Alternate proof of dmfex 7906: shorter but using ax-rep 5236. (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 7910	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 7911	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 7912	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 23488. (Contributed by NM, 1-Jan-2007.)
⊢ ¬ I ∈ V
Theorem	resiexg 7913	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7218). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
Theorem	imaexg 7914	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 7915	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 7916	The range of a set is a set. Deduction version of rnexd 7916. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran 𝐴 ∈ V)
Theorem	imaexd 7917	The image of a set is a set. Deduction version of imaexg 7914. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)
Theorem	exse2 7918	Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	xpexr 7919	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 7920	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 7921	A condition where the converse of xpex 7756 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 7922	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 7923	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7924, elxp6 8024, and elxp7 8025. (Contributed by NM, 17-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	elxp5 7924	Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7923 when the double intersection does not create class existence problems (caused by int0 4925). (Contributed by NM, 1-Aug-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	cnvexg 7925	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 7926	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 7927	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 7928	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 7929*	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 7930	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 7931	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V
Theorem	coexd 7932	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 7933*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6606 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)
Theorem	fun11uni 7934*	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 7935	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 7936	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 7937	A function with bounded domain and codomain is a set. This version of fex 7229 is proven without the Axiom of Replacement ax-rep 5236, but depends on ax-un 7740, which is not required for the proof of fex 7229. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
Theorem	fabexd 7938*	Existence of a set of functions. In contrast to fabex 7940 or fabexg 7939, 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 7939*	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	fabex 7940*	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 7941*	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 7942*	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	fiunlem 7943*	Lemma for fiun 7944 and f1iun 7945. Formerly part of f1iun 7945. (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 7944*	The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7945. (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 7945*	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 7946*	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 7947*	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 7948*	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 7949	Alternate proof of resfunexg 7218, shorter but requiring ax-pow 5334 and ax-un 7740. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)
Theorem	cofunexg 7950	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 7951	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 7952	Alternate proof of fnex 7220, derived using the Axiom of Replacement in the form of funimaexg 6623. This version uses ax-pow 5334 and ax-un 7740, whereas fnex 7220 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 7953	Weak version of funex 7222 that holds without ax-rep 5236. 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 7954*	Weak version of mptex 7226 that holds without ax-rep 5236. 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 7955	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 7222. (Contributed by NM, 11-Nov-1995.)
⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Theorem	zfrep6OLD 7956*	Obsolete proof of zfrep6 5248 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 7957	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))
Theorem	f1dmex 7958	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 5236. (Contributed by NM, 4-Sep-2004.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	f1ovv 7959	The codomain/range of a 1-1 onto function is a set iff its domain is a set. (Contributed by AV, 21-Mar-2019.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
Theorem	fvclex 7960*	Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
⊢ 𝐹 ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V
Theorem	fvresex 7961*	Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V
Theorem	abrexexg 7962*	Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5249, axrep6 5245, ax-rep 5236. See also abrexex2g 7965. There are partial converses under additional conditions, see for instance abnexg 7759. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2178, ax-11 2194, ax-12 2215, ax-pr 5402, ax-un 7740 and shorten proof. (Revised by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
Theorem	abrexex 7963*	Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7962. See also abrexex2 7970. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V
Theorem	iunexg 7964*	The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	abrexex2g 7965*	Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)
Theorem	opabex3d 7966*	Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 9-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)
Theorem	opabex3rd 7967*	Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.) (Revised by AV, 9-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ 𝜓} ∈ V)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ∈ V)
Theorem	opabex3 7968*	Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	iunex 7969*	The existence of an indexed union. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be read informally as 𝐵(𝑥). (Contributed by NM, 13-Oct-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V
Theorem	abrexex2 7970*	Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7963. (Contributed by NM, 12-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
Theorem	abexssex 7971*	Existence of a class abstraction with an existentially quantified expression. Both 𝑥 and 𝑦 can be free in 𝜑. (Contributed by NM, 29-Jul-2006.)
⊢ 𝐴 ∈ V    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V
Theorem	abexex 7972*	A condition where a class abstraction continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V
Theorem	f1oweALT 7973*	Alternate proof of one direction of f1owe 7358, more direct since not using the isomorphism predicate, but requiring ax-un 7740. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	wemoiso 7974*	Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 10120. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	wemoiso2 7975*	Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	oprabexd 7976*	Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by AV, 9-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓)    &   ⊢ (𝜑 → 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)})    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	oprabex 7977*	Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}    ⇒   ⊢ 𝐹 ∈ V
Theorem	oprabex3 7978*	Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
⊢ 𝐻 ∈ V    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}    ⇒   ⊢ 𝐹 ∈ V
Theorem	oprabrexex2 7979*	Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V
Theorem	ab2rexex 7980*	Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7963. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
Theorem	ab2rexex2 7981*	Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7970. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ {𝑧 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V
Theorem	xpexgALT 7982	Alternate proof of xpexg 7753 requiring Replacement (ax-rep 5236) but not Power Set (ax-pow 5334). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
Theorem	offval3 7983*	General value of (𝐹 ∘f 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
Theorem	offres 7984	Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷)))
Theorem	ofmres 7985*	Equivalent expressions for a restriction of the function operation map. Unlike ∘f 𝑅 which is a proper class, ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 7986, allowing it to be used as a function or structure argument. By ofmresval 7698, the restricted operation map values are the same as the original values, allowing theorems for ∘f 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)
⊢ ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘f 𝑅𝑔))
Theorem	ofmresex 7986	Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
Theorem	mptcnfimad 7987*	The converse of a mapping of subsets to their image of a bijection. (Contributed by AV, 23-Apr-2025.)
⊢ 𝑀 = (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)    &   ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))
2.4.8  First and second members of an ordered pair
Syntax	c1st 7988	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 7989	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 7990	Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 7998 proves that it does this. For example, (1st ‘⟨3, 4⟩) = 3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 6225 and op1stb 5451). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
Definition	df-2nd 7991	Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 7999 proves that it does this. For example, (2nd ‘⟨3, 4⟩) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6228 and op2ndb 6227). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
Theorem	1stval 7992	The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (1st ‘𝐴) = ∪ dom {𝐴}
Theorem	2ndval 7993	The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
Theorem	1stnpr 7994	Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅)
Theorem	2ndnpr 7995	Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)
Theorem	1st0 7996	The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (1st ‘∅) = ∅
Theorem	2nd0 7997	The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (2nd ‘∅) = ∅
Theorem	op1st 7998	Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Theorem	op2nd 7999	Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Theorem	op1std 8000	Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = 𝐴)