HomeHome Metamath Proof Explorer < Previous   Next >
Browser slow? Try the
Unicode version.

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 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21110

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-11900)
  Hilbert Space Explorer  Hilbert Space Explorer
(11901-13475)
  Users' Mathboxes  Users' Mathboxes
(13476-21110)
 

Statement List for Metamath Proof Explorer - 4101-4200 - Page 42 of 212   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtfinds2 4101* Transfinite Induction (inference schema), using implicit substititions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff ta is an auxiliary antecedent to help shorten proofs using this theorem.
|- (x = (/) -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- (ta -> ps)   &   |- (y e. On -> (ta -> (ch -> th)))   &   |- (Lim x -> (ta -> (A.y e. x ch -> ph)))   =>   |- (x e. On -> (ta -> ph))
 
Theoremtfinds3 4102* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. (Unnecessary distinct variable restrictions were removed by David Abernethy, 21-Jun-2011.)
|- (x = (/) -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- (et -> ps)   &   |- (y e. On -> (et -> (ch -> th)))   &   |- (Lim x -> (et -> (A.y e. x ch -> ph)))   =>   |- (A e. On -> (et -> ta))
 
The natural numbers (i.e. finite ordinals)
 
Syntaxcom 4103 Extend class notation to include the class of natural numbers.
class om
 
Definitiondf-om 4104* Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 4105 for an alternate definition. Later, when we assume the Axiom of Infinity, we show om is a set in omex 6139, and om can then be defined per dfom3 6144 (the smallest inductive set) and dfom4 6146.

Note: the natural numbers om are a subset of the ordinal numbers df-on 3833. They are completely different from the natural numbers NN (df-n 7769) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them.

|- om = {x | (Ord x /\ A.y(Lim y -> x e. y))}
 
Theoremdfom2 4105 An alternate definition of the set of natural numbers om. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 4090).
|- om = {x e. On | suc x C_ {y e. On | -. Lim y}}
 
Theoremelom 4106* Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 6145.
|- A e. _V   =>   |- (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x)))
 
Theoremelomg 4107* Membership in omega. The antecedent can be eliminated if we assume the Axiom of Infinity; see elom3 6145.
|- (A e. V -> (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x))))
 
Theoremomsson 4108 Omega is a subset of On. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- om C_ On
 
Theoremlimomss 4109 The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity.
|- (Lim A -> om C_ A)
 
Theoremnnon 4110 A natural number is an ordinal number.
|- (A e. om -> A e. On)
 
Theoremnnoni 4111 A natural number is an ordinal number.
|- A e. om   =>   |- A e. On
 
Theoremnnord 4112 A natural number is ordinal.
|- (A e. om -> Ord A)
 
Theoremordom 4113 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- Ord om
 
Theoremelnn 4114 A member of a natural number is a natural number.
|- ((A e. B /\ B e. om) -> A e. om)
 
Theoremomon 4115 The class of natural numbers om is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43.
|- (om e. On \/ om = On)
 
Theoremomelon2 4116 Omega is an ordinal number. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- (om e. _V -> om e. On)
 
Theoremnnlim 4117 A natural number is not a limit ordinal.
|- (A e. om -> -. Lim A)
 
Theoremomssnlim 4118 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- om C_ {x e. On | -. Lim x}
 
Theoremlimom 4119 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity.
|- Lim om
 
Theorempeano2b 4120 A class belongs to omega iff its successor does.
|- (A e. om <-> suc A e. om)
 
Theoremnnsuc 4121* A nonzero natural number is a successor.
|- ((A e. om /\ A =/= (/)) -> E.x e. om A = suc x)
 
Theoremssnlim 4122* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42.
|- ((Ord A /\ A C_ {x e. On | -. Lim x}) -> A C_ om)
 
Peano's postulates
 
Theorempeano1 4123 Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 4123 through peano5 4127 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity.
|- (/) e. om
 
Theorempeano2 4124 The successor of any natural number is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42.
|- (A e. om -> suc A e. om)
 
Theorempeano3 4125 The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42.
|- (A e. om -> suc A =/= (/))
 
Theorempeano4 4126 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43.
|- ((A e. om /\ B e. om) -> (suc A = suc B <-> A = B))
 
Theorempeano5 4127* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction hypothesis, is derived from this theorem as theorem findes 4134.
|- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
 
Theoremnn0suc 4128* A natural number is either 0 or a successor.
|- (A e. om -> (A = (/) \/ E.x e. om A = suc x))
 
Finite induction (for finite ordinals)
 
Theoremfind 4129* The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that A is a set of natural numbers, zero belongs to A, and given any member of A the member's successor also belongs to A. The conclusion is that every natural number is in A. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A C_ om /\ (/) e. A /\ A.x e. A suc x e. A)   =>   |- A = om
 
Theoremfinds 4130* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.
|- (x = (/) -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. om -> (ch -> th))   =>   |- (A e. om -> ta)
 
Theoremfindsg 4131* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. The basis of this version is an arbitrary natural number B instead of zero.
|- (x = B -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- (B e. om -> ps)   &   |- (((y e. om /\ B e. om) /\ B C_ y) -> (ch -> th))   =>   |- (((A e. om /\ B e. om) /\ B C_ A) -> ta)
 
Theoremfinds2 4132* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.
|- (x = (/) -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- (ta -> ps)   &   |- (y e. om -> (ta -> (ch -> th)))   =>   |- (x e. om -> (ta -> ph))
 
Theoremfinds1 4133* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.
|- (x = (/) -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = suc y -> (ph <-> th))   &   |- ps   &   |- (y e. om -> (ch -> th))   =>   |- (x e. om -> ph)
 
Theoremfindes 4134 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4100 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
|- [(/) / x]ph   &   |- (x e. om -> (ph -> [suc x / x]ph))   =>   |- (x e. om -> ph)
 
Functions and relations
 
Syntaxcxp 4135 Extend the definition of a class to include the cross product.
class (A X. B)
 
Syntaxccnv 4136 Extend the definition of a class to include the converse of a class.
class `'A
 
Syntaxcdm 4137 Extend the definition of a class to include the domain of a class.
class dom A
 
Syntaxcrn 4138 Extend the definition of a class to include the range of a class.
class ran A
 
Syntaxcres 4139 Extend the definition of a class to include the restriction of a class. (Read: The restriction of A to B.)
class (A |` B)
 
Syntaxcima 4140 Extend the definition of a class to include the image of a class. (Read: The image of B under A.)
class (A"B)
 
Syntaxccom 4141 Extend the definition of a class to include the composition of two classes. (Read: The composition of A and B.)
class (A o. B)
 
Syntaxwrel 4142 Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
 
Syntaxwfun 4143 Extend the definition of a wff to include the function predicate. (Read: A is a function.)
wff Fun A
 
Syntaxwfn 4144 Extend the definition of a wff to include the function predicate with a domain. (Read: A is a function on B.)
wff A Fn B
 
Syntaxwf 4145 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: F maps A into B.)
wff F:A-->B
 
Syntaxwf1 4146 Extend the definition of a wff to include one-to-one functions. (Read: F maps A one-to-one into B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
wff F:A-1-1->B
 
Syntaxwfo 4147 Extend the definition of a wff to include onto functions. (Read: F maps A onto B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
wff F:A-onto->B
 
Syntaxwf1o 4148 Extend the definition of a wff to include one-to-one onto functions. (Read: F maps A one-to-one onto B.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
wff F:A-1-1-onto->B
 
Syntaxcfv 4149 Extend the definition of a class to include the value of a function. (Read: The value of F at A, or "F of A.")
class (F` A)
 
Syntaxwiso 4150 Extend the definition of a wff to include the isomorphism property. (Read: H is an R, S isomorphism of A onto B.)
wff H Isom R, S (A, B)
 
Definitiondf-xp 4151* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64.
|- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
 
Definitiondf-rel 4152 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4509 and dfrel3 4531.
|- (Rel A <-> A C_ (_V X. _V))
 
Definitiondf-cnv 4153* Define the converse of a class. Definition 9.12 of [Quine] p. 64. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function.
|- `'A = {<.x, y>. | yAx}
 
Definitiondf-co 4154* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses A and B, uses /. instead of o., and calls the operation "relative product."
|- (A o. B) = {<.x, y>. | E.z(xBz /\ zAy)}
 
Definitiondf-dm 4155* Define the domain of a class. Definition 3 of [Suppes] p. 59. For alternate definitions see dfdm2 4573, dfdm3 4311, and dfdm4 4314. The notation "dom " is used by Enderton; other authors sometimes use script D.
|- dom A = {x | E.y xAy}
 
Definitiondf-rn 4156 Define the range of a class. For an alternate definitions, see dfrn2 4312, dfrn3 4313, and dfrn4 4534. The notation "ran " is used by Enderton; other authors sometimes use script R or script W.
|- ran A = dom `' A
 
Definitiondf-res 4157 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24.
|- (A |` B) = (A i^i (B X. _V))
 
Definitiondf-ima 4158 Define the image of a class. Definition 6.6(2) of [TakeutiZaring] p. 24. For an alternate definition, see dfima2 4425.
|- (A"B) = ran ( A |` B)
 
Definitiondf-fun 4159 Define a function. Definition 10.1 of [Quine] p. 65. For alternate definitions, see dffun2 4584, dffun3 4585, dffun4 4586, dffun5 4587, dffun6 4589, dffun7 4599, dffun8 4600, and dffun9 4601.
|- (Fun A <-> (Rel A /\ (A o. `'A) C_ _I ))
 
Definitiondf-fn 4160 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 4707, dffn3 4713, dffn4 4761, and dffn5 4856.
|- (A Fn B <-> (Fun A /\ dom A = B))
 
Definitiondf-f 4161 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 4944, dff3 4945, and dff4 4946.
|- (F:A-->B <-> (F Fn A /\ ran F C_ B))
 
Definitiondf-f1 4162 Define a one-to-one function. For equivalent definitions see dff12 4742 and dff13 5029. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow).
|- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
 
Definitiondf-fo 4163 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 4759, dffo3 4947, dffo4 4948, and dffo5 4949.
|- (F:A-onto->B <-> (F Fn A /\ ran F = B))
 
Definitiondf-f1o 4164 Define a one-to-one onto function. For equivalent definitions see dff1o2 4778, dff1o3 4779, dff1o4 4781, and dff1o5 4782. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow).
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
 
Definitiondf-fv 4165* Define the value of a function. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4451), our definition apparently does not appear in the literature; but it is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 4840 and fvprc 4813). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar F(A) notation for a function's value at A, i.e. "F of A," but without context-dependent notational ambiguity. Alternate definitions are dffv2 4873 and dffv3 5403. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 4812, fv3 4831, and fv4 5404. Restricted equivalents that require F to be a function are shown in funfv 4869 and funfv2 4870. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 4853.
|- (F` A) = U.{x | (F"{A}) = {x}}
 
Definitiondf-iso 4166* Define the isomorphism predicate. We read this as "H is an R, S isomorphism of A onto B." Normally, R and S are ordering relations on A and B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that R and S are subscripts.
|- (H Isom R, S (A, B) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))))
 
Theoremxpeq1 4167 Equality theorem for cross product.
|- (A = B -> (A X. C) = (B X. C))
 
Theoremxpeq2 4168 Equality theorem for cross product.
|- (A = B -> (C X. A) = (C X. B))
 
Theoremelxpi 4169* Membership in a cross product. Uses fewer axioms than elxp 4170.
|- (A e. (B X. C) -> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
 
Theoremelxp 4170* Membership in a cross product.
|- (A e. (B X. C) <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
 
Theoremelxp2 4171* Membership in a cross product.
|- (A e. (B X. C) <-> E.x e. B E.y e. C A = <.x, y>.)
 
Theoremxpeq12 4172 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
|- ((A = B /\ C = D) -> (A X. C) = (B X. D))
 
Theoremxpeq1i 4173 Equality inference for cross product.
|- A = B   =>   |- (A X. C) = (B X. C)
 
Theoremxpeq2i 4174 Equality inference for cross product.
|- A = B   =>   |- (C X. A) = (C X. B)
 
Theoremxpeq12i 4175 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
|- A = B   &   |- C = D   =>   |- (A X. C) = (B X. D)
 
Theoremxpeq1d 4176 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- (ph -> A = B)   =>   |- (ph -> (A X. C) = (B X. C))
 
Theoremxpeq2d 4177 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- (ph -> A = B)   =>   |- (ph -> (C X. A) = (C X. B))
 
Theoremxpeq12d 4178 Equality deduction for cross product.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A X. C) = (B X. D))
 
Theoremhbxp 4179* Bound-variable hypothesis builder for cross product. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A X. B) -> A.x y e. (A X. B))
 
Theoremcsbxpg 4180 Distribute proper substitution through the cross product of two classes. csbxpg 4180 is derived from the virtual deduction proof csbxpgVD 17342. (Contributed by Alan Sare, 10-Nov-2012. )
|- (A e. D -> [_A / x]_(B X. C) = ([_A / x]_B X. [_A / x]_C))
 
Theoremopelxp1 4181 The first member of an ordered pair of classes in a cross product belongs to first cross product argument.
|- (<.A, B>. e. (C X. D) -> A e. C)
 
Theoremotelxp1 4182 The first member of an ordered triple of classes in a cross product belongs to first cross product argument.
|- (<.<.A, B>., C>. e. ((R X. S) X. T) -> A e. R)
 
Theoremrabxp 4183* Membership in a class builder restricted to a cross product.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- {x e. (A X. B) | ph} = {<.y, z>. | (y e. A /\ z e. B /\ ps)}
 
Theorembrrelex 4184 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.)
|- ((Rel R /\ ARB) -> A e. _V)
 
Theorembrrelexi 4185 The first argument of a binary relation exists. (An artifact of our ordered pair definition.)
|- Rel R   =>   |- (ARB -> A e. _V)
 
Theoremnprrel 4186 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
|- Rel R   &   |- -. A e. _V   =>   |- -. ARB
 
Theoremfconstopab 4187* Representation of a constant function using ordered pairs.
|- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}
 
Theoremvtoclr 4188* Variable to class conversion of transitive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   =>   |- (C e. D -> ((ARB /\ BRC) -> ARC))
 
Theoremvtoclrbr 4189* Variable to class conversion of transitive, reflexive relation. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- xRx   =>   |- ((ARB /\ BRC) -> ARC)
 
Theoremvtoclibr 4190* Variable to class conversion of transitive, irreflexive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- -. xRx   =>   |- ((ARB /\ BRC) -> ARC)
 
Theoremopelxp 4191 Ordered pair membership in a cross product. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- B e. _V   =>   |- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))
 
Theorembrxp 4192 Binary relation on a cross product.
|- B e. _V   =>   |- (A(C X. D)B <-> (A e. C /\ B e. D))
 
Theoremopelxpg 4193 Ordered pair membership in a cross product.
|- (B e. V -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
 
Theoremopelxpi 4194 Ordered pair membership in a cross product (implication).
|- ((A e. C /\ B e. D) -> <.A, B>. e. (C X. D))
 
Theoremopelxpv 4195 Ordered pair membership in a semi-universal class of ordered pairs.
|- (A e. C <-> <.A, B>. e. (C X. _V))
 
Theoremopelvv 4196 Ordered pair membership in the universal class of ordered pairs.
|- A e. _V   =>   |- <.A, B>. e. (_V X. _V)
 
Theoremopprc4b 4197 A property of an ordered pair of proper classes (due to our particular definition of ordered pair).
|- (-. A e. _V <-> -. <.A, B>. e. (_V X. _V))
 
Theoremralxp 4198* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)
 
Theoremrexxp 4199* Existential quantification restricted to a cross product is equivalent to a double restricted quantification.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (E.x e. (A X. B)ph <-> E.y e. A E.z e. B ps)
 
Theoremralxpf 4200* Version of ralxp 4198 with bound-variable hypotheses.
|- (ph -> A.yph)   &   |- (ph -> A.zph)   &   |- (ps -> A.xps)   &   |- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)

MPE Home   Contents Copyright terms: Public domain < Previous  Next >