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	suppval1 7301*	The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍})
Theorem	suppvalfn 7302*	The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
Theorem	elsuppfn 7303	An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))
Theorem	cnvimadfsn 7304*	The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}
Theorem	suppimacnvss 7305	The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7296. (Contributed by AV, 7-Apr-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Theorem	suppimacnv 7306	Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = (◡𝑅 “ (V ∖ {𝑍})))
Theorem	frnsuppeq 7307	Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))
Theorem	suppssdm 7308	The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
Theorem	suppsnop 7309	The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
⊢ 𝐹 = {⟨𝑋, 𝑌⟩}    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))
Theorem	snopsuppss 7310	The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)
⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}
Theorem	fvn0elsupp 7311	If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
Theorem	fvn0elsuppb 7312	The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Theorem	rexsupp 7313*	Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (∃𝑥 ∈ (𝐹 supp 𝑍)𝜑 ↔ ∃𝑥 ∈ 𝑋 ((𝐹‘𝑥) ≠ 𝑍 ∧ 𝜑)))
Theorem	ressuppss 7314	The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	suppun 7315	The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
Theorem	ressuppssdif 7316	The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
Theorem	mptsuppdifd 7317*	The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})
Theorem	mptsuppd 7318*	The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍})
Theorem	extmptsuppeq 7319*	The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
Theorem	suppfnss 7320*	The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Theorem	funsssuppss 7321	The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
Theorem	fnsuppres 7322	Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹 ↾ 𝐵) = (𝐵 × {𝑍})))
Theorem	fnsuppeq0 7323	The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Theorem	fczsupp0 7324	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅
Theorem	suppss 7325*	Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
Theorem	suppssr 7326	A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)
Theorem	suppssov1 7327*	Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
Theorem	suppssof1 7328*	Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)    &   ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 ∘𝑓 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Theorem	suppss2 7329*	Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)
Theorem	suppsssn 7330*	Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ 𝑘 ≠ 𝑊) → 𝐵 = 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ {𝑊})
Theorem	suppssfv 7331*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)    &   ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) supp 𝑍) ⊆ 𝐿)
Theorem	suppofss1d 7332*	Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	suppofss2d 7333*	Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))
Theorem	supp0cosupp0 7334	The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))
Theorem	imacosupp 7335	The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))
2.4.9  Special "Maps to" operations The following theorems are about maps-to operations (see df-mpt2 6655) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 6733, ovmpt2x 6789 and fmpt2x 7236). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 7336*	Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5260. (Contributed by AV, 25-Oct-2020.)
⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	mpt2xeldm 7337*	If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
Theorem	mpt2xneldm 7338*	If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)
Theorem	mpt2xopn0yelv 7339*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
Theorem	mpt2xopynvov0g 7340*	If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
Theorem	mpt2xopxnop0 7341*	If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Theorem	mpt2xopx0ov0 7342*	If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (∅𝐹𝐾) = ∅
Theorem	mpt2xopxprcov0 7343*	If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
Theorem	mpt2xopynvov0 7344*	If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
Theorem	mpt2xopoveq 7345*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Theorem	mpt2xopovel 7346*	Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
Theorem	mpt2xopoveqd 7347*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    &   ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌))    &   ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)    ⇒   ⊢ (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Theorem	brovex 7348*	A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)    &   ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))    ⇒   ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
Theorem	brovmpt2ex 7349*	A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})    ⇒   ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
Theorem	sprmpt2d 7350*	The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)
⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)})    &   ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))    &   ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃))    &   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)    ⇒   ⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)})
2.4.10  Function transposition
Syntax	ctpos 7351	The transposition of a function.
class tpos 𝐹
Definition	df-tpos 7352*	Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
Theorem	tposss 7353	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Theorem	tposeq 7354	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Theorem	tposeqd 7355	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)
Theorem	tposssxp 7356	The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)
Theorem	reltpos 7357	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ Rel tpos 𝐹
Theorem	brtpos2 7358	Value of the transposition at a pair ⟨𝐴, 𝐵⟩. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
Theorem	brtpos0 7359	The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 7361. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
Theorem	reldmtpos 7360	Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Theorem	brtpos 7361	The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
Theorem	ottpos 7362	The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹))
Theorem	relbrtpos 7363	The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ (Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
Theorem	dmtpos 7364	The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
Theorem	rntpos 7365	The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Theorem	tposexg 7366	The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V)
Theorem	ovtpos 7367	The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to ℝ or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)
Theorem	tposfun 7368	The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Fun 𝐹 → Fun tpos 𝐹)
Theorem	dftpos2 7369*	Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
Theorem	dftpos3 7370*	Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5122. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})
Theorem	dftpos4 7371*	Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
Theorem	tpostpos 7372	Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Theorem	tpostpos2 7373	Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)
Theorem	tposfn2 7374	The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))
Theorem	tposfo2 7375	Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))
Theorem	tposf2 7376	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵))
Theorem	tposf12 7377	Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵))
Theorem	tposf1o2 7378	Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1-onto→𝐵 → tpos 𝐹:◡𝐴–1-1-onto→𝐵))
Theorem	tposfo 7379	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto→𝐶)
Theorem	tposf 7380	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)
Theorem	tposfn 7381	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))
Theorem	tpos0 7382	Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
⊢ tpos ∅ = ∅
Theorem	tposco 7383	Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺)
Theorem	tpossym 7384*	Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
Theorem	tposeqi 7385	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ tpos 𝐹 = tpos 𝐺
Theorem	tposex 7386	A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 ∈ V    ⇒   ⊢ tpos 𝐹 ∈ V
Theorem	nftpos 7387	Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥tpos 𝐹
Theorem	tposoprab 7388*	Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}    ⇒   ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
Theorem	tposmpt2 7389*	Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ tpos 𝐹 = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
Theorem	tposconst 7390	The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.)
⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})
2.4.11  Curry and uncurry
Syntax	ccur 7391	Extend class notation to include the currying function.
class curry 𝐴
Syntax	cunc 7392	Extend class notation to include the uncurrying function.
class uncurry 𝐴
Definition	df-cur 7393*	Define the currying of 𝐹, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
Definition	df-unc 7394*	Define the uncurrying of 𝐹, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
Theorem	mpt2curryd 7395*	The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
Theorem	mpt2curryvald 7396*	The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶))
Theorem	fvmpt2curryd 7397*	The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))
2.4.12  Undefined values
Syntax	cund 7398	Extend class notation with undefined value function.
class Undef
Definition	df-undef 7399	Define the undefined value function, whose value at set 𝑠 is guaranteed not to be a member of 𝑠 (see pwuninel 7401). (Contributed by NM, 15-Sep-2011.)
⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
Theorem	pwuninel2 7400	Direct proof of pwuninel 7401 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)