HomeHome Intuitionistic Logic Explorer
Theorem List (p. 45 of 85)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopabbi2dv 4401* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2130. (Contributed by NM, 24-Feb-2014.)
Rel A    &   (φ → (⟨x, y Aψ))       (φA = {⟨x, y⟩ ∣ ψ})
 
Theoremrelop 4402* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
A V    &   B V       (Rel ⟨A, B⟩ ↔ xy(A = {x} B = {x, y}))
 
Theoremideqg 4403 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(B 𝑉 → (A I BA = B))
 
Theoremideq 4404 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
B V       (A I BA = B)
 
Theoremididg 4405 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A 𝑉A I A)
 
Theoremissetid 4406 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(A V ↔ A I A)
 
Theoremcoss1 4407 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(AB → (A𝐶) ⊆ (B𝐶))
 
Theoremcoss2 4408 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(AB → (𝐶A) ⊆ (𝐶B))
 
Theoremcoeq1 4409 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremcoeq2 4410 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremcoeq1i 4411 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (A𝐶) = (B𝐶)
 
Theoremcoeq2i 4412 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (𝐶A) = (𝐶B)
 
Theoremcoeq1d 4413 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremcoeq2d 4414 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremcoeq12i 4415 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)
 
Theoremcoeq12d 4416 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremnfco 4417 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
xA    &   xB       x(AB)
 
Theorembrcog 4418* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
 
Theoremopelco2g 4419* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (⟨A, B (𝐶𝐷) ↔ x(⟨A, x 𝐷 x, B 𝐶)))
 
Theorembrcogw 4420 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A(𝐶𝐷)B)
 
Theoremeqbrrdva 4421* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(φA ⊆ (𝐶 × 𝐷))    &   (φB ⊆ (𝐶 × 𝐷))    &   ((φ x 𝐶 y 𝐷) → (xAyxBy))       (φA = B)
 
Theorembrco 4422* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (A(𝐶𝐷)Bx(A𝐷x x𝐶B))
 
Theoremopelco 4423* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (⟨A, B (𝐶𝐷) ↔ x(A𝐷x x𝐶B))
 
Theoremcnvss 4424 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
(ABAB)
 
Theoremcnveq 4425 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(A = BA = B)
 
Theoremcnveqi 4426 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
A = B       A = B
 
Theoremcnveqd 4427 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(φA = B)       (φA = B)
 
Theoremelcnv 4428* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(A 𝑅xy(A = ⟨x, y y𝑅x))
 
Theoremelcnv2 4429* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(A 𝑅xy(A = ⟨x, yy, x 𝑅))
 
Theoremnfcnv 4430 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xA
 
Theoremopelcnvg 4431 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))
 
Theorembrcnvg 4432 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))
 
Theoremopelcnv 4433 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (⟨A, B 𝑅 ↔ ⟨B, A 𝑅)
 
Theorembrcnv 4434 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (A𝑅BB𝑅A)
 
Theoremcsbcnvg 4435 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
(A 𝑉A / x𝐹 = A / x𝐹)
 
Theoremcnvco 4436 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = (BA)
 
Theoremcnvuni 4437* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
A = x A x
 
Theoremdfdm3 4438* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
dom A = {xyx, y A}
 
Theoremdfrn2 4439* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran A = {yx xAy}
 
Theoremdfrn3 4440* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
ran A = {yxx, y A}
 
Theoremelrn2g 4441* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bxx, A B))
 
Theoremelrng 4442* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bx xBA))
 
Theoremdfdm4 4443 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom A = ran A
 
Theoremdfdmf 4444* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       dom A = {xy xAy}
 
Theoremcsbdmg 4445 Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
(A 𝑉A / xdom B = dom A / xB)
 
Theoremeldmg 4446* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom By ABy))
 
Theoremeldm2g 4447* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom ByA, y B))
 
Theoremeldm 4448* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
A V       (A dom By ABy)
 
Theoremeldm2 4449* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
A V       (A dom ByA, y B)
 
Theoremdmss 4450 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(AB → dom A ⊆ dom B)
 
Theoremdmeq 4451 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(A = B → dom A = dom B)
 
Theoremdmeqi 4452 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
A = B       dom A = dom B
 
Theoremdmeqd 4453 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → dom A = dom B)
 
Theoremopeldm 4454 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (⟨A, B 𝐶A dom 𝐶)
 
Theorembreldm 4455 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (A𝑅BA dom 𝑅)
 
Theoremopeldmg 4456 Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
((A 𝑉 B 𝑊) → (⟨A, B 𝐶A dom 𝐶))
 
Theorembreldmg 4457 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)
 
Theoremdmun 4458 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (AB) = (dom A ∪ dom B)
 
Theoremdmin 4459 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
dom (AB) ⊆ (dom A ∩ dom B)
 
Theoremdmiun 4460 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom x A B = x A dom B
 
Theoremdmuni 4461* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
dom A = x A dom x
 
Theoremdmopab 4462* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
dom {⟨x, y⟩ ∣ φ} = {xyφ}
 
Theoremdmopabss 4463* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨x, y⟩ ∣ (x A φ)} ⊆ A
 
Theoremdmopab3 4464* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
 
Theoremdm0 4465 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom ∅ = ∅
 
Theoremdmi 4466 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom I = V
 
Theoremdmv 4467 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V
 
Theoremdm0rn0 4468 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
(dom A = ∅ ↔ ran A = ∅)
 
Theoremreldm0 4469 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel A → (A = ∅ ↔ dom A = ∅))
 
Theoremdmmrnm 4470* A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
(x x dom Ay y ran A)
 
Theoremdmxpm 4471* The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(x x B → dom (A × B) = A)
 
Theoremdmxpinm 4472* The domain of the intersection of two square cross products. Unlike dmin 4459, equality holds. (Contributed by NM, 29-Jan-2008.)
(x x (AB) → dom ((A × A) ∩ (B × B)) = (AB))
 
Theoremxpid11m 4473* The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
((x x A x x B) → ((A × A) = (B × B) ↔ A = B))
 
Theoremdmcnvcnv 4474 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4687). (Contributed by NM, 8-Apr-2007.)
dom A = dom A
 
Theoremrncnvcnv 4475 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
ran A = ran A
 
Theoremelreldm 4476 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
((Rel A B A) → B dom A)
 
Theoremrneq 4477 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(A = B → ran A = ran B)
 
Theoremrneqi 4478 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
A = B       ran A = ran B
 
Theoremrneqd 4479 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → ran A = ran B)
 
Theoremrnss 4480 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(AB → ran A ⊆ ran B)
 
Theorembrelrng 4481 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((A 𝐹 B 𝐺 A𝐶B) → B ran 𝐶)
 
Theoremopelrng 4482 Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
((A 𝐹 B 𝐺 A, B 𝐶) → B ran 𝐶)
 
Theorembrelrn 4483 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
A V    &   B V       (A𝐶BB ran 𝐶)
 
Theoremopelrn 4484 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
A V    &   B V       (⟨A, B 𝐶B ran 𝐶)
 
Theoremreleldm 4485 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → A dom 𝑅)
 
Theoremrelelrn 4486 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → B ran 𝑅)
 
Theoremreleldmb 4487* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A dom 𝑅x A𝑅x))
 
Theoremrelelrnb 4488* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A ran 𝑅x x𝑅A))
 
Theoremreleldmi 4489 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (A𝑅BA dom 𝑅)
 
Theoremrelelrni 4490 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (A𝑅BB ran 𝑅)
 
Theoremdfrnf 4491* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       ran A = {yx xAy}
 
Theoremelrn2 4492* Membership in a range. (Contributed by NM, 10-Jul-1994.)
A V       (A ran Bxx, A B)
 
Theoremelrn 4493* Membership in a range. (Contributed by NM, 2-Apr-2004.)
A V       (A ran Bx xBA)
 
Theoremnfdm 4494 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xdom A
 
Theoremnfrn 4495 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xran A
 
Theoremdmiin 4496 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom x A B x A dom B
 
Theoremrnopab 4497* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
ran {⟨x, y⟩ ∣ φ} = {yxφ}
 
Theoremrnmpt 4498* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)       ran 𝐹 = {yx A y = B}
 
Theoremelrnmpt 4499* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
𝐹 = (x AB)       (𝐶 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
 
Theoremelrnmpt1s 4500* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (x AB)    &   (x = 𝐷B = 𝐶)       ((𝐷 A 𝐶 𝑉) → 𝐶 ran 𝐹)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8449
  Copyright terms: Public domain < Previous  Next >