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	elsetpreimafvbi 47401*	An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
Theorem	elsetpreimafveqfv 47402*	The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐹‘𝑋) = (𝐹‘𝑌))
Theorem	eqfvelsetpreimafv 47403*	If an element of the domain of the function has the same function value as an element of the preimage of a function value, then it is an element of the same preimage. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)) → 𝑌 ∈ 𝑆))
Theorem	elsetpreimafvrab 47404*	An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})
Theorem	imaelsetpreimafv 47405*	The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
Theorem	uniimaelsetpreimafv 47406*	The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
Theorem	elsetpreimafveq 47407*	If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑆 = 𝑅))
Theorem	fundcmpsurinjlem1 47408*	Lemma 1 for fundcmpsurinj 47419. (Contributed by AV, 4-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ⊢ ran 𝐺 = 𝑃
Theorem	fundcmpsurinjlem2 47409*	Lemma 2 for fundcmpsurinj 47419. (Contributed by AV, 4-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃)
Theorem	fundcmpsurinjlem3 47410*	Lemma 3 for fundcmpsurinj 47419. (Contributed by AV, 3-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))
Theorem	imasetpreimafvbijlemf 47411*	Lemma for imasetpreimafvbij 47416: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
Theorem	imasetpreimafvbijlemfv 47412*	Lemma for imasetpreimafvbij 47416: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))
Theorem	imasetpreimafvbijlemfv1 47413*	Lemma for imasetpreimafvbij 47416: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))
Theorem	imasetpreimafvbijlemf1 47414*	Lemma for imasetpreimafvbij 47416: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))
Theorem	imasetpreimafvbijlemfo 47415*	Lemma for imasetpreimafvbij 47416: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))
Theorem	imasetpreimafvbij 47416*	The mapping 𝐻 is a bijective function between the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–1-1-onto→(𝐹 “ 𝐴))
Theorem	fundcmpsurbijinjpreimafv 47417*	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 47418*	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 47419*	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 47420*	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 47421*	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 47422*	Alternate proof of fundcmpsurinj 47419, based on fundcmpsurinjimaid 47421: 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 47423	Extend class notation with the partitions of a closed interval of extended reals.
class RePart
Definition	df-iccp 47424*	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 47425*	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 47426*	A special partition. Corresponds to fourierdlem2 46126 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 47427	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 47428	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 47429	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 47430	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 47431	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 47432*	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 47433*	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 47434	If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 46135 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
Theorem	iccpartltu 47435*	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 47436*	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 47437*	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 47438*	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 47439*	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 47440	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 47441	The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 46139 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 47442	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 47443*	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 47444*	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 47445	Lemma for icceuelpart 47446. (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 47446*	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 47447*	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 47448	A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 46136 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 47449*	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 47450*	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 47451*	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 47452*	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 47453*	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 47454	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 47455	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 47456*	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 47464. Another, equivalent definition using two temporary setvar variables is provided in dfich2 47468. (Contributed by AV, 29-Jul-2023.)
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
Theorem	nfich1 47457	The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑
Theorem	nfich2 47458	The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑
Theorem	ichv 47459*	Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]𝜑
Theorem	ichf 47460	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 47461	A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑥]𝜑
Theorem	icht 47462	A theorem is interchangeable. (Contributed by SN, 4-May-2024.)
⊢ 𝜑    ⇒   ⊢ [𝑥⇄𝑦]𝜑
Theorem	ichbidv 47463*	Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑥⇄𝑦]𝜓 ↔ [𝑥⇄𝑦]𝜒))
Theorem	ichcircshi 47464*	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 47465	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 47456 instead of dfich2 47468 to reduce axioms. (Revised by SN, 4-May-2024.)
⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))
Theorem	ichn 47466	Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑)
Theorem	ichim 47467	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 47468*	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 47469*	The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)
⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓)
Theorem	ichbi12i 47470*	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 47471	In an equality for the same setvar variable, the setvar variable is interchangeable by itself. Special case of ichid 47461 and icheq 47472 without distinct variables restriction. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑥]𝑥 = 𝑥
Theorem	icheq 47472*	In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.)
⊢ [𝑥⇄𝑦]𝑥 = 𝑦
Theorem	ichnfimlem 47473*	Lemma for ichnfim 47474: A substitution for a nonfree variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2371. (Revised by GG, 1-May-2024.)
⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
Theorem	ichnfim 47474*	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 47475*	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 47476*	Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)
⊢ (∀𝑥[𝑎⇄𝑏]𝜑 → [𝑎⇄𝑏]∀𝑥𝜑)
Theorem	ich2al 47477	Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]∀𝑥∀𝑦𝜑
Theorem	ich2ex 47478	Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑
Theorem	ichexmpl1 47479*	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 47480*	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 47481*	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 47482*	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 47483*	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 47484	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 47485*	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 4718, which is not an element of all unordered pairs, see spr0nelg 47486. (Contributed by AV, 21-Nov-2021.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
Theorem	spr0nelg 47486*	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 47487	Extend class notation with set of pairs.
class Pairs
Definition	df-spr 47488*	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 47489*	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 47490*	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 47491*	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 47492	The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ ∅ ∉ (Pairs‘𝑉)
Theorem	sprvalpwn0 47493*	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 47494*	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 47495*	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 47496	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 47497	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 47498	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 47499*	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 47500*	Lemma for sprsymrelf 47505 and sprsymrelfv 47504. (Contributed by AV, 19-Nov-2021.)
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))