P. 74 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	dff14b 7301*	A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦)))
Theorem	dff15 7302*	A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)))
Theorem	f1resveqaeq 7303	If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (((𝐹 ↾ 𝐴):𝐴⟶1-1𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	f1resrcmplf1dlem 7304	Lemma for f1resrcmplf1d 7305. (Contributed by BTernaryTau, 27-Sep-2023.) (Revised by Mingli Yuan, 15-Aug-2026.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ (𝜑 → (𝐹‘𝑋) = (𝐹‘𝑌))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	f1resrcmplf1d 7305	If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)⟶1-1𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶1-1𝐵)
Theorem	f1ounsn 7306	Extension of a bijection by an ordered pair. (Contributed by AV, 15-Sep-2025.)
⊢ 𝐹 = (𝐺 ∪ {⟨𝑋, 𝑌⟩})    ⇒   ⊢ ((𝐺:𝐴⟶1-1onto𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝐹:(𝐴 ∪ {𝑋})⟶1-1onto(𝐵 ∪ {𝑌}))
Theorem	f12dfv 7307	A one-to-one function with a domain with at least two different elements in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
⊢ 𝐴 = {𝑋, 𝑌}    ⇒   ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹‘𝑋) ≠ (𝐹‘𝑌))))
Theorem	f13dfv 7308	A one-to-one function with a domain with at least three different elements in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
⊢ 𝐴 = {𝑋, 𝑌, 𝑍}    ⇒   ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) ∧ (𝐹‘𝑋) ≠ (𝐹‘𝑍) ∧ (𝐹‘𝑌) ≠ (𝐹‘𝑍)))))
Theorem	dff1o6 7309*	A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
⊢ (𝐹:𝐴⟶1-1onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
Theorem	f1ocnvfv1 7310	The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv2 7311	The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv 7312	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶))
Theorem	f1ocnvfvb 7313	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶))
Theorem	nvof1o 7314	An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶1-1onto𝐴)
Theorem	nvocnv 7315*	The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)
⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)
Theorem	f1cdmsn 7316*	If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ((𝐹:𝐴⟶1-1{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
Theorem	fsnex 7317*	Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)
⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓))
Theorem	f1prex 7318*	Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.)
⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = (𝑓‘𝐵) → (𝜒 ↔ 𝜑))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}⟶1-1𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝜓)))
Theorem	f1ocnvdm 7319	The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴)
Theorem	f1ocnvfvrneq 7320	If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	fcof1 7321	An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴⟶1-1𝐵)
Theorem	fcofo 7322	An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶onto𝐵)
Theorem	cbvfo 7323*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹:𝐴⟶onto𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
Theorem	cbvexfo 7324*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹:𝐴⟶onto𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	cocan1 7325	An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))
Theorem	cocan2 7326	A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))
Theorem	fcof1oinvd 7327	Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 7330. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))    ⇒   ⊢ (𝜑 → ◡𝐹 = 𝐺)
Theorem	fcof1od 7328	A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7321 and fcofo 7322. Formerly part of proof of fcof1o 7330. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))    &   ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐵)
Theorem	2fcoidinvd 7329	Show that a function is the inverse of a function if their compositions are the identity functions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))    &   ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))    ⇒   ⊢ (𝜑 → ◡𝐹 = 𝐺)
Theorem	fcof1o 7330	Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.)
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴⟶1-1onto𝐵 ∧ ◡𝐹 = 𝐺))
Theorem	2fvcoidd 7331*	Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
Theorem	2fvidf1od 7332*	A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐵)
Theorem	2fvidinvd 7333*	Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)    ⇒   ⊢ (𝜑 → ◡𝐹 = 𝐺)
Theorem	foeqcnvco 7334	Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
Theorem	f1eqcocnv 7335	Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Proof shortened by Wolf Lammen, 29-May-2024.)
⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐺:𝐴⟶1-1𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
Theorem	fveqf1o 7336	Given a bijection 𝐹, produce another bijection 𝐺 which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))    ⇒   ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺:𝐴⟶1-1onto𝐵 ∧ (𝐺‘𝐶) = 𝐷))
Theorem	f1ocoima 7337	The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵))
Theorem	nf1const 7338	A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.)
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴⟶1-1𝐶)
Theorem	nf1oconst 7339	A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.)
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴⟶1-1onto𝐶)
Theorem	f1ofvswap 7340	Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴⟶1-1onto𝐵)
Theorem	fvf1pr 7341	Values of a one-to-one function between two sets with two elements. Actually, such a function is a bijection. (Contributed by AV, 22-Jul-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}⟶1-1{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))
Theorem	fliftrel 7342*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
Theorem	fliftel 7343*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
Theorem	fliftel1 7344*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵)
Theorem	fliftcnv 7345*	Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
Theorem	fliftfun 7346*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷)))
Theorem	fliftfund 7347*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 = 𝐶)) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fliftfuns 7348*	The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
Theorem	fliftf 7349*	The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))
Theorem	fliftval 7350*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)
Theorem	isoeq1 7351	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq2 7352	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq3 7353	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))
Theorem	isoeq4 7354	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
Theorem	isoeq5 7355	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))
Theorem	nfiso 7356	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎ𝑥𝐻    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Theorem	isof1o 7357	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴⟶1-1onto𝐵)
Theorem	isof1oidb 7358	A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)
⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))
Theorem	isof1oopb 7359	A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.)
⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
Theorem	isorel 7360	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	soisores 7361*	Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))
Theorem	soisoi 7362*	Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴⟶onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	isoid 7363	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Theorem	isocnv 7364	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
Theorem	isocnv2 7365	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
Theorem	isocnv3 7366	Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)    &   ⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)    ⇒   ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))
Theorem	isores2 7367	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
Theorem	isores1 7368	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
Theorem	isores3 7369	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Theorem	isotr 7370	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))
Theorem	isomin 7371	Isomorphisms preserve minimal elements. Note that (◡𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥 ∣ 𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
Theorem	isoini 7372	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
Theorem	isoini2 7373	Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))    &   ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))    ⇒   ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Theorem	isofrlem 7374*	Lemma for isofr 7376. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	isoselem 7375*	Lemma for isose 7377. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
Theorem	isofr 7376	An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
Theorem	isose 7377	An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))
Theorem	isofr2 7378	A weak form of isofr 7376 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	isopolem 7379	Lemma for isopo 7380. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
Theorem	isopo 7380	An isomorphism preserves the property of being a partial order. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))
Theorem	isosolem 7381	Lemma for isoso 7382. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	isoso 7382	An isomorphism preserves the property of being a strict total order. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
Theorem	isowe 7383	An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
Theorem	isowe2 7384*	A weak form of isowe 7383 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	f1oiso 7385*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
⊢ ((𝐻:𝐴⟶1-1onto𝐵 ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	f1oiso2 7386*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))}    ⇒   ⊢ (𝐻:𝐴⟶1-1onto𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	f1owe 7387*	Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.) Strengthen to biconditional. (Revised by Eric Schmidt, 5-Aug-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
Theorem	f1oweOLD 7388*	Obsolete version of f1owe as of 5-Aug-2026. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	f1we 7389*	Pull back a well ordering by a one-to-one function. (Contributed by Eric Schmidt, 4-Aug-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   ⊢ (𝐹:𝐴⟶1-1𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))
Theorem	weniso 7390	A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 Isom 𝑅, 𝑅 (𝐴, 𝐴)) → 𝐹 = ( I ↾ 𝐴))
Theorem	weisoeq 7391	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 8012. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	weisoeq2 7392	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 8013. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	knatar 7393*	The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)
⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))
Theorem	fvresval 7394	The value of a restricted function at a class is either the empty set or the value of the unrestricted function at that class. (Contributed by Scott Fenton, 4-Sep-2011.)
⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
Theorem	funeldmb 7395	If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))
Theorem	eqfunresadj 7396	Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))
Theorem	eqfunressuc 7397	Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))
Theorem	fnssintima 7398*	Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
Theorem	fnimasnd 7399	The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})
2.3.17  Cantor's Theorem
Theorem	canth 7400	No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 9180. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7401 for a counterexample. (Use nex 1833 if you want the form ¬ ∃𝑓𝑓:𝐴⟶onto𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐹:𝐴⟶onto𝒫 𝐴