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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfeq23 5101 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
 
Theoremfeq1d 5102 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
 
Theoremfeq2d 5103 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
 
Theoremfeq12d 5104 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
 
Theoremfeq123d 5105 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
 
Theoremfeq123 5106 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
 
Theoremfeq1i 5107 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:𝐴𝐵𝐺:𝐴𝐵)
 
Theoremfeq2i 5108 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
𝐴 = 𝐵       (𝐹:𝐴𝐶𝐹:𝐵𝐶)
 
Theoremfeq23i 5109 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝐹:𝐴𝐵𝐹:𝐶𝐷)
 
Theoremfeq23d 5110 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
 
Theoremnff 5111 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴𝐵
 
Theoremsbcfng 5112* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
 
Theoremsbcfg 5113* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
 
Theoremffn 5114 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵𝐹 Fn 𝐴)
 
Theoremffnd 5115 A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 Fn 𝐴)
 
Theoremdffn2 5116 Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn 𝐴𝐹:𝐴⟶V)
 
Theoremffun 5117 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Fun 𝐹)
 
Theoremfrel 5118 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Rel 𝐹)
 
Theoremfdm 5119 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
 
Theoremfdmi 5120 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴
 
Theoremfrn 5121 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)
 
Theoremdffn3 5122 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
 
Theoremfss 5123 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
 
Theoremfssd 5124 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremfco 5125 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfco2 5126 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
(((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfssxp 5127 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
 
Theoremfex2 5128 A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
 
Theoremfunssxp 5129 Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremffdm 5130 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremopelf 5131 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
 
Theoremfun 5132 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
 
Theoremfun2 5133 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 
Theoremfnfco 5134 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
 
Theoremfssres 5135 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfssres2 5136 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfresin 5137 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
 
Theoremresasplitss 5138 If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))
 
Theoremfcoi1 5139 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
 
Theoremfcoi2 5140 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
 
Theoremfeu 5141* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
 
Theoremfcnvres 5142 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
 
Theoremfimacnvdisj 5143 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfintm 5144* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
𝑥 𝑥𝐵       (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
 
Theoremfin 5145 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
 
Theoremfabexg 5146* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremfabex 5147* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       𝐹 ∈ V
 
Theoremdmfex 5148 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
 
Theoremf0 5149 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶𝐴
 
Theoremf00 5150 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
(𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf0bi 5151 A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
(𝐹:∅⟶𝑋𝐹 = ∅)
 
Theoremf0dom0 5152 A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
(𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
 
Theoremf0rn0 5153* If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
 
Theoremfconst 5154 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
𝐵 ∈ V       (𝐴 × {𝐵}):𝐴⟶{𝐵}
 
Theoremfconstg 5155 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
 
Theoremfnconstg 5156 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(𝐵𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
 
Theoremfconst6g 5157 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
 
Theoremfconst6 5158 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝐵𝐶       (𝐴 × {𝐵}):𝐴𝐶
 
Theoremf1eq1 5159 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵))
 
Theoremf1eq2 5160 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
 
Theoremf1eq3 5161 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
 
Theoremnff1 5162 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1𝐵
 
Theoremdff12 5163* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
 
Theoremf1f 5164 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
 
Theoremf1rn 5165 The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
(𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
 
Theoremf1fn 5166 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
 
Theoremf1fun 5167 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Fun 𝐹)
 
Theoremf1rel 5168 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Rel 𝐹)
 
Theoremf1dm 5169 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
 
Theoremf1ss 5170 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssr 5171 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ff1 5172 If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)
((𝐹:𝐴1-1𝐵𝐹:𝐴𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssres 5173 A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
 
Theoremf1resf1 5174 The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
(((𝐹:𝐴1-1𝐵𝐶𝐴) ∧ (𝐹𝐶):𝐶𝐷) → (𝐹𝐶):𝐶1-1𝐷)
 
Theoremf1cnvcnv 5175 Two ways to express that a set 𝐴 (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
(𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
 
Theoremf1co 5176 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
 
Theoremfoeq1 5177 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 
Theoremfoeq2 5178 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
 
Theoremfoeq3 5179 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
 
Theoremnffo 5180 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴onto𝐵
 
Theoremfof 5181 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
 
Theoremfofun 5182 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴onto𝐵 → Fun 𝐹)
 
Theoremfofn 5183 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
 
Theoremforn 5184 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
 
Theoremdffo2 5185 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremfoima 5186 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 
Theoremdffn4 5187 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
 
Theoremfunforn 5188 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴𝐴:dom 𝐴onto→ran 𝐴)
 
Theoremfodmrnu 5189 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremfores 5190 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
 
Theoremfoco 5191 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 
Theoremf1oeq1 5192 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
 
Theoremf1oeq2 5193 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
 
Theoremf1oeq3 5194 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 
Theoremf1oeq23 5195 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
 
Theoremf1eq123d 5196 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))
 
Theoremfoeq123d 5197 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
 
Theoremf1oeq123d 5198 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
 
Theoremnff1o 5199 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1-onto𝐵
 
Theoremf1of1 5200 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
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