P. 475 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47401-47500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fundcmpsurbijinjpreimafv 47401*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjpreimafv 47402*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurinj 47403*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurbijinj 47404*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjimaid 47405*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto the image (𝐹 “ 𝐴) of the domain of 𝐹 and an injective function from the image (𝐹 “ 𝐴). (Contributed by AV, 17-Mar-2024.)
⊢ 𝐼 = (𝐹 “ 𝐴)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))    &   ⊢ 𝐻 = ( I ↾ 𝐼)    ⇒   ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))
Theorem	fundcmpsurinjALT 47406*	Alternate proof of fundcmpsurinj 47403, based on fundcmpsurinjimaid 47405: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
21.48.8  Partitions of real intervals Based on the theorems of the fourierdlem* series of GS's mathbox.
Syntax	ciccp 47407	Extend class notation with the partitions of a closed interval of extended reals.
class RePart
Definition	df-iccp 47408*	Define partitions of a closed interval of extended reals. Such partitions are finite increasing sequences of extended reals. (Contributed by AV, 8-Jul-2020.)
⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
Theorem	iccpval 47409*	Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)
⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
Theorem	iccpart 47410*	A special partition. Corresponds to fourierdlem2 46100 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
Theorem	iccpartimp 47411	Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
Theorem	iccpartres 47412	The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
Theorem	iccpartxr 47413	If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)
Theorem	iccpartgtprec 47414	If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼))
Theorem	iccpartipre 47415	If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ)
Theorem	iccpartiltu 47416*	If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
Theorem	iccpartigtl 47417*	If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
Theorem	iccpartlt 47418	If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 46109 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
Theorem	iccpartltu 47419*	If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
Theorem	iccpartgtl 47420*	If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃‘𝑖))
Theorem	iccpartgt 47421*	If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))
Theorem	iccpartleu 47422*	If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀))
Theorem	iccpartgel 47423*	If there is a partition, then all intermediate points and the upper and the lower bound are greater than or equal to the lower bound. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑖))
Theorem	iccpartrn 47424	If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))
Theorem	iccpartf 47425	The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 46113 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → 𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀)))
Theorem	iccpartel 47426	If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ ((𝜑 ∧ 𝐼 ∈ (0...𝑀)) → (𝑃‘𝐼) ∈ ((𝑃‘0)[,](𝑃‘𝑀)))
Theorem	iccelpart 47427*	An element of any partitioned half-open interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
⊢ (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
Theorem	iccpartiun 47428*	A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ((𝑃‘0)[,)(𝑃‘𝑀)) = ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
Theorem	icceuelpartlem 47429	Lemma for icceuelpart 47430. (Contributed by AV, 19-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 < 𝐽 → (𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽))))
Theorem	icceuelpart 47430*	An element of a partitioned half-open interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
Theorem	iccpartdisj 47431*	The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → Disj 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
Theorem	iccpartnel 47432	A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 46110 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝑃)    ⇒   ⊢ ((𝜑 ∧ 𝐼 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑃‘𝐼)(,)(𝑃‘(𝐼 + 1))))
21.48.9  Shifting functions with an integer range domain
Theorem	fargshiftfv 47433*	If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1))))
Theorem	fargshiftf 47434*	If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
Theorem	fargshiftf1 47435*	If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸)
Theorem	fargshiftfo 47436*	If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸)
Theorem	fargshiftfva 47437*	The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
21.48.10  Words over a set (extension)
21.48.10.1  Last symbol of a word - extension
Theorem	lswn0 47438	The last symbol of a nonempty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases (∅ is the last symbol) and invalid cases (∅ means that no last symbol exists). This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (♯‘𝑊) ≠ 0) → (lastS‘𝑊) ≠ ∅)
21.48.11  Unordered pairs
21.48.11.1  Interchangeable setvar variables
Syntax	wich 47439	Extend wff notation to include the property of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. Read this notation as "𝑥 and 𝑦 are interchangeable in wff 𝜑".
wff [𝑥⇄𝑦]𝜑
Definition	df-ich 47440*	Define the property of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. For an alternate definition using implicit substitution and a temporary setvar variable see ichcircshi 47448. Another, equivalent definition using two temporary setvar variables is provided in dfich2 47452. (Contributed by AV, 29-Jul-2023.)
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
Theorem	nfich1 47441	The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑
Theorem	nfich2 47442	The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑
Theorem	ichv 47443*	Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]𝜑
Theorem	ichf 47444	Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ [𝑥⇄𝑦]𝜑
Theorem	ichid 47445	A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑥]𝜑
Theorem	icht 47446	A theorem is interchangeable. (Contributed by SN, 4-May-2024.)
⊢ 𝜑    ⇒   ⊢ [𝑥⇄𝑦]𝜑
Theorem	ichbidv 47447*	Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑥⇄𝑦]𝜓 ↔ [𝑥⇄𝑦]𝜒))
Theorem	ichcircshi 47448*	The setvar variables are interchangeable if they can be circularily shifted using a third setvar variable, using implicit substitution. (Contributed by AV, 29-Jul-2023.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   ⊢ [𝑥⇄𝑦]𝜑
Theorem	ichan 47449	If two setvar variables are interchangeable in two wffs, then they are interchangeable in the conjunction of these two wffs. Notice that the reverse implication is not necessarily true. Corresponding theorems will hold for other commutative operations, too. (Contributed by AV, 31-Jul-2023.) Use df-ich 47440 instead of dfich2 47452 to reduce axioms. (Revised by SN, 4-May-2024.)
⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))
Theorem	ichn 47450	Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑)
Theorem	ichim 47451	Formula building rule for implication in interchangeability. (Contributed by SN, 4-May-2024.)
⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 → 𝜓))
Theorem	dfich2 47452*	Alternate definition of the property of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. (Contributed by AV and WL, 6-Aug-2023.)
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
Theorem	ichcom 47453*	The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)
⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓)
Theorem	ichbi12i 47454*	Equivalence for interchangeable setvar variables. (Contributed by AV, 29-Jul-2023.)
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑎⇄𝑏]𝜒)
Theorem	icheqid 47455	In an equality for the same setvar variable, the setvar variable is interchangeable by itself. Special case of ichid 47445 and icheq 47456 without distinct variables restriction. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑥]𝑥 = 𝑥
Theorem	icheq 47456*	In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑦]𝑥 = 𝑦
Theorem	ichnfimlem 47457*	Lemma for ichnfim 47458: A substitution for a nonfree variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2370. (Revised by GG, 1-May-2024.)
⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
Theorem	ichnfim 47458*	If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)
Theorem	ichnfb 47459*	If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑))
Theorem	ichal 47460*	Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)
⊢ (∀𝑥[𝑎⇄𝑏]𝜑 → [𝑎⇄𝑏]∀𝑥𝜑)
Theorem	ich2al 47461	Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]∀𝑥∀𝑦𝜑
Theorem	ich2ex 47462	Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑
Theorem	ichexmpl1 47463*	Example for interchangeable setvar variables in a statement of predicate calculus with equality. (Contributed by AV, 31-Jul-2023.)
⊢ [𝑎⇄𝑏]∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)
Theorem	ichexmpl2 47464*	Example for interchangeable setvar variables in an arithmetic expression. (Contributed by AV, 31-Jul-2023.)
⊢ [𝑎⇄𝑏]((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝑎↑2) + (𝑏↑2)) = (𝑐↑2))
Theorem	ich2exprop 47465*	If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
Theorem	ichnreuop 47466*	If the setvar variables are interchangeable in a wff, there is never a unique ordered pair with different components fulfilling the wff (because if ⟨𝑎, 𝑏⟩ fulfils the wff, then also ⟨𝑏, 𝑎⟩ fulfils the wff). (Contributed by AV, 27-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 → ¬ ∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑎 ≠ 𝑏 ∧ 𝜑))
Theorem	ichreuopeq 47467*	If the setvar variables are interchangeable in a wff, and there is a unique ordered pair fulfilling the wff, then both setvar variables must be equal. (Contributed by AV, 28-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
21.48.11.2  Set of unordered pairs
Theorem	sprid 47468	Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
Theorem	elsprel 47469*	An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4719, which is not an element of all unordered pairs, see spr0nelg 47470. (Contributed by AV, 21-Nov-2021.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
Theorem	spr0nelg 47470*	The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
Syntax	cspr 47471	Extend class notation with set of pairs.
class Pairs
Definition	df-spr 47472*	Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
Theorem	sprval 47473*	The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
Theorem	sprvalpw 47474*	The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
Theorem	sprssspr 47475*	The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
Theorem	spr0el 47476	The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ ∅ ∉ (Pairs‘𝑉)
Theorem	sprvalpwn0 47477*	The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
Theorem	sprel 47478*	An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})
Theorem	prssspr 47479*	An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋 ∈ 𝑃) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})
Theorem	prelspr 47480	An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Theorem	prsprel 47481	The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
Theorem	prsssprel 47482	The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
Theorem	sprvalpwle2 47483*	The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Theorem	sprsymrelfvlem 47484*	Lemma for sprsymrelf 47489 and sprsymrelfv 47488. (Contributed by AV, 19-Nov-2021.)
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
Theorem	sprsymrelf1lem 47485*	Lemma for sprsymrelf1 47490. (Contributed by AV, 22-Nov-2021.)
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏))
Theorem	sprsymrelfolem1 47486*	Lemma 1 for sprsymrelfo 47491. (Contributed by AV, 22-Nov-2021.)
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}    ⇒   ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)
Theorem	sprsymrelfolem2 47487*	Lemma 2 for sprsymrelfo 47491. (Contributed by AV, 23-Nov-2021.)
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}))
Theorem	sprsymrelfv 47488*	The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})    ⇒   ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}})
Theorem	sprsymrelf 47489*	The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})    ⇒   ⊢ 𝐹:𝑃⟶𝑅
Theorem	sprsymrelf1 47490*	The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})    ⇒   ⊢ 𝐹:𝑃–1-1→𝑅
Theorem	sprsymrelfo 47491*	The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅)
Theorem	sprsymrelf1o 47492*	The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅)
Theorem	sprbisymrel 47493*	There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅)
Theorem	sprsymrelen 47494*	The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅)
21.48.11.3  Proper (unordered) pairs Proper (unordered) pairs are unordered pairs with exactly 2 elements. The set of proper pairs with elements of a class 𝑉 is defined by {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}. For example, {1, 2} is a proper pair, because 1 ≠ 2 ( see 1ne2 12331). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4813), {1, V} = {1} (see prprc2 4718) or {V, V} = ∅ (see prprc 4719).
Theorem	prpair 47495*	Characterization of a proper pair: A class is a proper pair iff it consists of exactly two different sets. (Contributed by AV, 11-Mar-2023.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
Theorem	prproropf1olem0 47496	Lemma 0 for prproropf1o 47501. Remark: 𝑂, the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, can alternatively be written as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ (1st ‘𝑥)𝑅(2nd ‘𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.)
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    ⇒   ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊)))
Theorem	prproropf1olem1 47497*	Lemma 1 for prproropf1o 47501. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    ⇒   ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃)
Theorem	prproropf1olem2 47498*	Lemma 2 for prproropf1o 47501. (Contributed by AV, 13-Mar-2023.)
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    ⇒   ⊢ ((𝑅 Or 𝑉 ∧ 𝑋 ∈ 𝑃) → ⟨inf(𝑋, 𝑉, 𝑅), sup(𝑋, 𝑉, 𝑅)⟩ ∈ 𝑂)
Theorem	prproropf1olem3 47499*	Lemma 3 for prproropf1o 47501. (Contributed by AV, 13-Mar-2023.)
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)    ⇒   ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → (𝐹‘{(1st ‘𝑊), (2nd ‘𝑊)}) = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
Theorem	prproropf1olem4 47500*	Lemma 4 for prproropf1o 47501. (Contributed by AV, 14-Mar-2023.)
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)    ⇒   ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ((𝐹‘𝑍) = (𝐹‘𝑊) → 𝑍 = 𝑊))