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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunoprabg 6801* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})

Theoremfunoprab 6802* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
∃*𝑧𝜑       Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Theoremfnoprabg 6803* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})

Theoremmpt2fun 6804* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Fun 𝐹

Theoremfnoprab 6805* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
(𝜑 → ∃!𝑧𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theoremffnov 6806* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))

Theoremfovcl 6807 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremeqfnov 6808* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))

Theoremeqfnov2 6809* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))

Theoremfnov 6810* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))

Theoremmpt22eqb 6811* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6809. (Contributed by Mario Carneiro, 4-Jan-2017.)
(∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))

Theoremrnmpt2 6812* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}

Theoremreldmmpt2 6813* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Rel dom 𝐹

Theoremelrnmpt2g 6814* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))

Theoremelrnmpt2 6815* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)

Theoremelrnmpt2res 6816* Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝐷𝑉 → (𝐷 ∈ ran (𝐹𝑅) ↔ ∃𝑥𝐴𝑦𝐵 (𝐷 = 𝐶𝑥𝑅𝑦)))

Theoremralrnmpt2 6817* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))

Theoremrexrnmpt2 6818* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))

Theoremovid 6819* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))

Theoremovidig 6820* The value of an operation class abstraction. Compare ovidi 6821. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
∃*𝑧𝜑    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       (𝜑 → (𝑥𝐹𝑦) = 𝑧)

Theoremovidi 6821* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝑥𝑅𝑦𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))

Theoremov 6822* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶𝜃))

Theoremovigg 6823* The value of an operation class abstraction. Compare ovig 6824. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   ∃*𝑧𝜑    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Theoremovig 6824* The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Theoremovmpt4g 6825* Value of a function given by the "maps to" notation. (This is the operation analogue of fvmpt2 6330.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)

Theoremovmpt2s 6826* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)

Theoremov2gf 6827* The value of an operation class abstraction. A version of ovmpt2g 6837 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐺    &   𝑦𝑆    &   (𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2dxf 6828* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐿)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2dx 6829* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐿)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2d 6830* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2x 6831* The value of an operation class abstraction. Variant of ovmpt2ga 6832 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑥 = 𝐴𝐷 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2ga 6832* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2a 6833* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   𝑆 ∈ V       ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2df 6834* Alternate deduction version of ovmpt2 6838, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))    &   𝑥𝐹    &   𝑥𝜓    &   𝑦𝐹    &   𝑦𝜓       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))

Theoremovmpt2dv 6835* Alternate deduction version of ovmpt2 6838, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))

Theoremovmpt2dv2 6836* Alternate deduction version of ovmpt2 6838, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))

Theoremovmpt2g 6837* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2 6838* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   𝑆 ∈ V       ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)

Theoremov3 6839* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
𝑆 ∈ V    &   (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}       (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆)

Theoremov6g 6840* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
(⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)}       (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐴𝐹𝐵) = 𝑆)

Theoremovg 6841* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝜏 ∧ (𝑥𝑅𝑦𝑆)) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → ((𝐴𝐹𝐵) = 𝐶𝜃))

Theoremovres 6842 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
((𝐴𝐶𝐵𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))

Theoremovresd 6843 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Theoremoprres 6844* The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.)
((𝜑𝑥𝑌𝑦𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    &   (𝜑𝑌𝑋)    &   (𝜑𝐹:(𝑌 × 𝑌)⟶𝑅)    &   (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)       (𝜑𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Theoremoprssov 6845 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
(((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Theoremfovrn 6846 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremfovrnda 6847 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremfovrnd 6848 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑆)       (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremfnrnov 6849* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)
(𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})

Theoremfoov 6850* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
(𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))

Theoremfnovrn 6851 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)

Theoremovelrn 6852* A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
(𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))

Theoremfunimassov 6853* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))

Theoremovelimab 6854* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦)))

Theoremovima0 6855 An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)
((𝑋𝐴𝑌𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))

Theoremovconst2 6856 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
𝐶 ∈ V       ((𝑅𝐴𝑆𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶)

Theoremoprssdm 6857* Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)
¬ ∅ ∈ 𝑆    &   ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝑆 × 𝑆) ⊆ dom 𝐹

Theoremnssdmovg 6858 The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)
((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Theoremndmovg 6859 The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Theoremndmov 6860 The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)

Theoremndmovcl 6861 The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
dom 𝐹 = (𝑆 × 𝑆)    &   ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)    &   ∅ ∈ 𝑆       (𝐴𝐹𝐵) ∈ 𝑆

Theoremndmovrcl 6862 Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
dom 𝐹 = (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆       ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))

Theoremndmovcom 6863 Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Theoremndmovass 6864 Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
dom 𝐹 = (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Theoremndmovdistr 6865 Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
dom 𝐹 = (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆    &   dom 𝐺 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

Theoremndmovord 6866 Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
dom 𝐹 = (𝑆 × 𝑆)    &   𝑅 ⊆ (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆    &   ((𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremndmovordi 6867 Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)
dom 𝐹 = (𝑆 × 𝑆)    &   𝑅 ⊆ (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆    &   (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))       ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

Theoremcaovclg 6868* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)       ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)

Theoremcaovcld 6869* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)

Theoremcaovcl 6870* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)       ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)

Theoremcaovcomg 6871* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Theoremcaovcomd 6872* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Theoremcaovcom 6873* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Theoremcaovassg 6874* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Theoremcaovassd 6875* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Theoremcaovass 6876* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

Theoremcaovcang 6877* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))       ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

Theoremcaovcand 6878* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

Theoremcaovcanrd 6879* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))

Theoremcaovcan 6880* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
𝐶 ∈ V    &   ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))       ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))

Theoremcaovordig 6881* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremcaovordid 6882* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremcaovordg 6883* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremcaovordd 6884* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremcaovord2d 6885* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Theoremcaovord3d 6886* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐷𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶𝐷𝑅𝐵)))

Theoremcaovord 6887* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Theoremcaovord2 6888* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Theoremcaovord3 6889* Ordering law. (Contributed by NM, 29-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   𝐷 ∈ V       (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))

Theoremcaovdig 6890* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))       ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

Theoremcaovdid 6891* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

Theoremcaovdir2d 6892* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))

Theoremcaovdirg 6893* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Theoremcaovdird 6894* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Theoremcaovdi 6895* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))       (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))

Theoremcaov32d 6896* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵))

Theoremcaov12d 6897* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))

Theoremcaov31d 6898* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴))

Theoremcaov13d 6899* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴)))

Theoremcaov4d 6900* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐷𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))

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