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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1oen2g 8501 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8503 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Theoremf1dom2g 8502 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8504 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Theoremf1oeng 8503 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Theoremf1domg 8504 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
(𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))

Theoremf1oen 8505 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
𝐴 ∈ V       (𝐹:𝐴1-1-onto𝐵𝐴𝐵)

Theoremf1dom 8506 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
𝐵 ∈ V       (𝐹:𝐴1-1𝐵𝐴𝐵)

Theorembrsdom 8507 Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))

Theoremisfi 8508* Express "𝐴 is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
(𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)

Theoremenssdom 8509 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
≈ ⊆ ≼

Theoremdfdom2 8510 Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
≼ = ( ≺ ∪ ≈ )

Theoremendom 8511 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
(𝐴𝐵𝐴𝐵)

Theoremsdomdom 8512 Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
(𝐴𝐵𝐴𝐵)

Theoremsdomnen 8513 Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
(𝐴𝐵 → ¬ 𝐴𝐵)

Theorembrdom2 8514 Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))

Theorembren2 8515 Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))

Theoremenrefg 8516 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉𝐴𝐴)

Theoremenref 8517 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴 ∈ V       𝐴𝐴

Theoremeqeng 8518 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
(𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))

Theoremdomrefg 8519 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
(𝐴𝑉𝐴𝐴)

Theoremen2d 8520* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶 ∈ V))    &   (𝜑 → (𝑦𝐵𝐷 ∈ V))    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑𝐴𝐵)

Theoremen3d 8521* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → (𝑦𝐵𝐷𝐴))    &   (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))       (𝜑𝐴𝐵)

Theoremen2i 8522* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶 ∈ V)    &   (𝑦𝐵𝐷 ∈ V)    &   ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))       𝐴𝐵

Theoremen3i 8523* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶𝐵)    &   (𝑦𝐵𝐷𝐴)    &   ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))       𝐴𝐵

Theoremdom2lem 8524* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)

Theoremdom2d 8525* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝐵𝑅𝐴𝐵))

Theoremdom3d 8526* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑𝐴𝐵)

Theoremdom2 8527* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
(𝑥𝐴𝐶𝐵)    &   ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))       (𝐵𝑉𝐴𝐵)

Theoremdom3 8528* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
(𝑥𝐴𝐶𝐵)    &   ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))       ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)

Theoremidssen 8529 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
I ⊆ ≈

Theoremssdomg 8530 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝐵𝑉 → (𝐴𝐵𝐴𝐵))

Theoremener 8531 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 1-May-2021.)
≈ Er V

Theoremensymb 8532 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)

Theoremensym 8533 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)

Theoremensymi 8534 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵       𝐵𝐴

Theoremensymd 8535 Symmetry of equinumerosity. Deduction form of ensym 8533. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)

Theorementr 8536 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremdomtr 8537 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theorementri 8538 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶

Theorementr2i 8539 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐶𝐴

Theorementr3i 8540 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐴𝐶       𝐵𝐶

Theorementr4i 8541 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐶𝐵       𝐴𝐶

Theoremendomtr 8542 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremdomentr 8543 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremf1imaeng 8544 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐹:𝐴1-1𝐵𝐶𝐴𝐶𝑉) → (𝐹𝐶) ≈ 𝐶)

Theoremf1imaen2g 8545 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 8546 does not need ax-reg 9032.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
(((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)

Theoremf1imaen 8546 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
𝐶 ∈ V       ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) ≈ 𝐶)

Theoremen0 8547 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Theoremensn1 8548 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
𝐴 ∈ V       {𝐴} ≈ 1o

Theoremensn1g 8549 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → {𝐴} ≈ 1o)

Theoremenpr1g 8550 {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
(𝐴𝑉 → {𝐴, 𝐴} ≈ 1o)

Theoremen1 8551* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
(𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Theoremen1b 8552 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
(𝐴 ≈ 1o𝐴 = { 𝐴})

Theoremreuen1 8553* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1o)

Theoremeuen1 8554 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1o)

Theoremeuen1b 8555* Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴 ≈ 1o ↔ ∃!𝑥 𝑥𝐴)

Theoremen1uniel 8556 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(𝑆 ≈ 1o 𝑆𝑆)

Theorem2dom 8557* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
(2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)

Theoremfundmen 8558 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐹 ∈ V       (Fun 𝐹 → dom 𝐹𝐹)

Theoremfundmeng 8559 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)

Theoremcnven 8560 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
((Rel 𝐴𝐴𝑉) → 𝐴𝐴)

Theoremcnvct 8561 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → 𝐴 ≼ ω)

Theoremfndmeng 8562 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)

Theoremmapsnend 8563 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴m {𝐵}) ≈ 𝐴)

Theoremmapsnen 8564 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 17-Jul-2022.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴m {𝐵}) ≈ 𝐴

Theoremsnmapen 8565 Set exponentiation: a singleton to any set is equinumerous to that singleton. (Contributed by NM, 17-Dec-2003.) (Revised by AV, 17-Jul-2022.)
((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Theoremsnmapen1 8566 Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.)
((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o)

Theoremmap1 8567 Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.)
(𝐴𝑉 → (1om 𝐴) ≈ 1o)

Theoremen2sn 8568 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})

Theoremsnfi 8569 A singleton is finite. (Contributed by NM, 4-Nov-2002.)
{𝐴} ∈ Fin

Theoremfiprc 8570 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Fin ∉ V

Theoremunen 8571 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((𝐴𝐵𝐶𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) ≈ (𝐵𝐷))

Theoremenpr2d 8572 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐵} ≈ 2o)

Theoremssct 8573 Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)

Theoremdifsnen 8574 All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.)
((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

Theoremdomdifsn 8575 Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))

Theoremxpsnen 8576 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × {𝐵}) ≈ 𝐴

Theoremxpsneng 8577 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Theoremxp1en 8578 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 1o) ≈ 𝐴)

Theoremendisj 8579* Any two sets are equinumerous to two disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)

Theoremundom 8580 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼ (𝐵𝐷))

Theoremxpcomf1o 8581* The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})       𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)

Theoremxpcomco 8582* Composition with the bijection of xpcomf1o 8581 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})    &   𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)       (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)

Theoremxpcomen 8583 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

Theoremxpcomeng 8584 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Theoremxpsnen2g 8585 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)

Theoremxpassen 8586 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))

Theoremxpdom2 8587 Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐶 ∈ V       (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Theoremxpdom2g 8588 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Theoremxpdom1g 8589 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))

Theoremxpdom3 8590 A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))

Theoremxpdom1 8591 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
𝐶 ∈ V       (𝐴𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))

Theoremdomunsncan 8592 A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))

Theoremomxpenlem 8593* Lemma for omxpen 8594. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)
𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))

Theoremomxpen 8594 The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ≈ (𝐴 × 𝐵))

Theoremomf1o 8595* Construct an explicit bijection from 𝐴 ·o 𝐵 to 𝐵 ·o 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))    &   𝐺 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Theorempw2f1olem 8596* Lemma for pw2f1o 8597. (Contributed by Mario Carneiro, 6-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑊)    &   (𝜑𝐵𝐶)       (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶}))))

Theorempw2f1o 8597* The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑊)    &   (𝜑𝐵𝐶)    &   𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)))       (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑m 𝐴))

Theorempw2eng 8598 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2o. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
(𝐴𝑉 → 𝒫 𝐴 ≈ (2om 𝐴))

Theorempw2en 8599 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
𝐴 ∈ V       𝒫 𝐴 ≈ (2om 𝐴)

Theoremfopwdom 8600 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.)
((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

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