P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	problem3 36401	Practice problem 3. Clues: eqcomi 2770 eqtri 2784 subaddrii 11628 recni 11304 4re 12408 3re 12404 1re 11289 df-4 12388 addcomi 11482. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 36402	Practice problem 4. Clues: pm3.2i 476 eqcomi 2770 eqtri 2784 subaddrii 11628 recni 11304 7re 12417 6re 12414 ax-1cn 11239 df-7 12391 ax-mp 5 oveq1i 7422 3cn 12405 2cn 12399 df-3 12387 mullidi 11295 subdiri 11747 mp3an 1490 mulcli 11297 subadd23 11550 oveq2i 7423 oveq12i 7424 3t2e6 12489 mulcomi 11298 subcli 11615 biimpri 231 subadd2i 11627. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 3    &   ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7    ⇒   ⊢ (𝐴 = 1 ∧ 𝐵 = 2)
Theorem	problem5 36403	Practice problem 5. Clues: 3brtr3i 5134 mpbi 233 breqtri 5130 ltaddsubi 11858 remulcli 11306 2re 12398 3re 12404 9re 12423 eqcomi 2770 mvlladdi 11557 3cn 6cn 12415 eqtr3i 2786 6p3e9 12483 addcomi 11482 ltdiv1ii 12227 6re 12414 nngt0i 12358 2nn 12397 divcan3i 12044 recni 11304 2cn 12399 2ne0 12430 mpbir 234 eqtri 2784 mulcomi 11298 3t2e6 12489 divmuli 12052. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 36404	Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
⊢ 𝑋 ∈ ℂ    &   ⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0    ⇒   ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))
21.8  Mathbox for Paul Chapman
21.8.1  Real and complex numbers (cont.)
Theorem	climuzcnv 36405*	Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑))))
Theorem	sinccvglem 36406*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))    &   ⊢ (𝜑 → 𝐹 ⇝ 0)    &   ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))    &   ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)
Theorem	sinccvg 36407*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1)
Theorem	circum 36408*	The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ 𝐴 = ((2 · π) / 𝑛)    &   ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))))    &   ⊢ 𝑅 ∈ ℝ    ⇒   ⊢ 𝑃 ⇝ ((2 · π) · 𝑅)
21.8.2  Miscellaneous theorems
Theorem	elfzm12 36409	Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁)))
Theorem	nn0seqcvg 36410*	A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ 𝐹:ℕ0⟶ℕ0    &   ⊢ 𝑁 = (𝐹‘0)    &   ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘)))    ⇒   ⊢ (𝐹‘𝑁) = 0
Theorem	lediv2aALT 36411	Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	abs2sqlei 36412	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))
Theorem	abs2sqlti 36413	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))
Theorem	abs2sqle 36414	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)))
Theorem	abs2sqlt 36415	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)))
Theorem	abs2difi 36416	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 36417	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))
21.9  Mathbox for Hongxiu Chen
Theorem	2thALT 36418	Alternate proof of 2th 267. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	orbi2iALT 36419	Alternate proof of orbi2i 926. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
Theorem	pm3.48ALT 36420	Alternate proof of pm3.48 978. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))
Theorem	3jcadALT 36421	Alternate proof of 3jcad 1147. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) Use 3jcad instead. (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))
21.10  Mathbox for Adrian Ducourtial
21.10.1  Propositional calculus
Theorem	currybi 36422	Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 37399 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)
⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)
Theorem	antnest 36423	Suppose 𝜑, 𝜓 are distinct atomic propositional formulas, and let Γ be the smallest class of formulas for which ⊤ ∈ Γ and (𝜒 → 𝜑), (𝜒 → 𝜓) ∈ Γ for 𝜒 ∈ Γ. The present theorem is then an element of Γ, and the implications occurring in the theorem are in one-to-one correspondence with the formulas in Γ up to logical equivalence. In particular, the theorem itself is equivalent to ⊤ ∈ Γ. (Contributed by Adrian Ducourtial, 2-Oct-2025.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
Theorem	antnestlaw3lem 36424	Lemma for antnestlaw3 36427. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ (¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestlaw1 36425	A law of nested antecedents. The converse direction is a subschema of pm2.27 43. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))
Theorem	antnestlaw2 36426	A law of nested antecedents. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))
Theorem	antnestlaw3 36427	A law of nested antecedents. Compare with looinv 206. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestALT 36428	Alternative proof of antnest 36423 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 36425 and antnestlaw3 36427. (Contributed by Adrian Ducourtial, 5-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
21.10.2  Clone theory
Syntax	ccloneop 36429	Syntax for the function of the class of operations on a set.
class CloneOp
Definition	df-cloneop 36430*	Define the function that sends a set to the class of clone-theoretic operations on the set. For convenience, we take an operation on 𝑎 to be a function on finite sequences of elements of 𝑎 (rather than tuples) with values in 𝑎. Following line 6 of [Szendrei] p. 11, the arity 𝑛 of an operation (here, the length of the sequences at which the operation is defined) is always finite and nonzero, whence 𝑛 is taken to be a nonzero finite ordinal. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
Syntax	cprj 36431	Syntax for the function of projections on sets.
class prj
Definition	df-prj 36432*	Define the function that, for a set 𝑎, arity 𝑛, and index 𝑖, returns the 𝑖-th 𝑛-ary projection on 𝑎. This is the 𝑛-ary operation on 𝑎 that, for any sequence of 𝑛 elements of 𝑎, returns the element having index 𝑖. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
Syntax	csuppos 36433	Syntax for the function of superpositions.
class suppos
Definition	df-suppos 36434*	Define the function that, when given an 𝑛-ary operation 𝑓 and 𝑛 many 𝑚-ary operations (𝑔‘∅), ..., (𝑔‘∪ 𝑛), returns the superposition of 𝑓 with the (𝑔‘𝑖), itself another 𝑚-ary operation on 𝑎. Given 𝑥 (a sequence of 𝑚 arguments in 𝑎), the superposition effectively applies each of the (𝑔‘𝑖) to 𝑥, then applies 𝑓 to the resulting sequence of 𝑛 function values. This can be seen as a generalized version of function composition; see paragraph 3 of [Szendrei] p. 11. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (𝑓 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛)), 𝑔 ∈ ((𝑎 ↑m (𝑎 ↑m 𝑚)) ↑m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
21.11  Mathbox for Scott Fenton
21.11.1  ZFC Axioms in primitive form
Theorem	axextprim 36435	ax-ext 2733 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))
Theorem	axrepprim 36436	ax-rep 5232 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥)))
Theorem	axunprim 36437	ax-un 7740 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpowprim 36438	ax-pow 5327 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦)
Theorem	axregprim 36439	ax-reg 9570 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axinfprim 36440	ax-inf 9623 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥))))
Theorem	axacprim 36441	ax-ac 10518 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
21.11.2  Untangled classes
Theorem	untelirr 36442*	We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 36524). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
Theorem	untuni 36443*	The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)
Theorem	untsucf 36444*	If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	unt0 36445	The empty set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥
Theorem	untint 36446*	If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	efrunt 36447*	If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
Theorem	untangtr 36448*	A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
21.11.3  Extra propositional calculus theorems
Theorem	3jaodd 36449	Double deduction form of 3jaoi 1454. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂)))
Theorem	3orit 36450	Closed form of 3ori 1451. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	biimpexp 36451	A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))
21.11.4  Misc. Useful Theorems
Theorem	nepss 36452	Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))
Theorem	3ccased 36453	Triple disjunction form of ccased 1054. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓))    ⇒   ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓))
Theorem	dfso3 36454*	Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	brtpid1 36455	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵
Theorem	brtpid2 36456	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵
Theorem	brtpid3 36457	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵
Theorem	iota5f 36458*	A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	jath 36459	Closed form of ja 188. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.)
⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒)))
Theorem	xpab 36460*	Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
Theorem	nnuni 36461	The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
21.11.5  Properties of real and complex numbers
Theorem	sqdivzi 36462	Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
Theorem	supfz 36463	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)
Theorem	inffz 36464	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
Theorem	fz0n 36465	The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0))
Theorem	shftvalg 36466	Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴)))
Theorem	divcnvlin 36467*	Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵)))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
Theorem	climlec3 36468*	Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	iexpire 36469	i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.)
⊢ (i↑𝑐i) ∈ ℝ
Theorem	bcneg1 36470	The binomial coefficient over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0)
Theorem	bcm1nt 36471	The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾))))
Theorem	bcprod 36472*	A product identity for binomial coefficients. (Contributed by Scott Fenton, 23-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))
Theorem	bccolsum 36473*	A column-sum rule for binomial coefficients. (Contributed by Scott Fenton, 24-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))
21.11.6  Infinite products
Theorem	iprodefisumlem 36474	Lemma for iprodefisum 36475. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Theorem	iprodefisum 36475*	Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))
Theorem	iprodgam 36476*	An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
21.11.7  Factorial limits
Theorem	faclimlem1 36477*	Lemma for faclim 36480. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1))))))
Theorem	faclimlem2 36478*	Lemma for faclim 36480. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1))
Theorem	faclimlem3 36479	Lemma for faclim 36480. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵)))))
Theorem	faclim 36480*	An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))    ⇒   ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))
Theorem	iprodfac 36481*	An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))))
Theorem	faclim2 36482*	Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))    ⇒   ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1)
21.11.8  Greatest common divisor and divisibility
Theorem	gcd32 36483	Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵))
Theorem	gcdabsorb 36484	Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵))
21.11.9  Properties of relationships
Theorem	dftr6 36485	A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Theorem	coep 36486*	Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)
Theorem	coepr 36487*	Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)
Theorem	dffr5 36488	A quantifier-free definition of a well-founded relation. (Contributed by Scott Fenton, 11-Apr-2011.) (Proof shortened by Scott Fenton, 26-Aug-2026.)
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))
Theorem	dfso2 36489	Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))
Theorem	br8 36490*	Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))    &   ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}    ⇒   ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))
Theorem	br6 36491*	Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))
Theorem	br4 36492*	Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏))
Theorem	cnvco1 36493	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)
Theorem	cnvco2 36494	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)
Theorem	eldm3 36495	Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Theorem	elrn3 36496	Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
Theorem	pocnv 36497	The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)
Theorem	socnv 36498	The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴)
Theorem	elintfv 36499*	Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ 𝑋 ∈ V    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
21.11.10  Properties of functions and mappings
Theorem	funpsstri 36500	A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))