HomeTheorem List (p. 367 of 512)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | faclimlem1 36601* | Lemma for faclim 36604. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1)))))) | ||
| Theorem | faclimlem2 36602* | Lemma for faclim 36604. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1)) | ||
| Theorem | faclimlem3 36603 | Lemma for faclim 36604. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵))))) | ||
| Theorem | faclim 36604* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) ⇒ ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴)) | ||
| Theorem | iprodfac 36605* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) | ||
| Theorem | faclim2 36606* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀)))) ⇒ ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1) | ||
| Theorem | gcd32 36607 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵)) | ||
| Theorem | gcdabsorb 36608 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵)) | ||
| Theorem | dftr6 36609 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) | ||
| Theorem | coep 36610* | Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) | ||
| Theorem | coepr 36611* | Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) | ||
| Theorem | dffr5 36612 | A quantifier-free definition of a well-founded relation. (Contributed by Scott Fenton, 11-Apr-2011.) (Proof shortened by Scott Fenton, 26-Aug-2026.) |
| ⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅))) | ||
| Theorem | dfso2 36613 | Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
| ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)))) | ||
| Theorem | br8 36614* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌)) | ||
| Theorem | br6 36615* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁)) | ||
| Theorem | br4 36616* | Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏)) | ||
| Theorem | cnvco1 36617 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | ||
| Theorem | cnvco2 36618 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) | ||
| Theorem | eldm3 36619 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅) | ||
| Theorem | elrn3 36620 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) | ||
| Theorem | pocnv 36621 | The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) | ||
| Theorem | socnv 36622 | The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) | ||
| Theorem | elintfv 36623* | Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ 𝑋 ∈ V ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | ||
| Theorem | funpsstri 36624 | A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.) |
| ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹)) | ||
| Theorem | fundmpss 36625 | If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.) |
| ⊢ (Fun 𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺)) | ||
| Theorem | funsseq 36626 | Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺)) | ||
| Theorem | funbreq 36627 | An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) | ||
| Theorem | br1steq 36628 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴) | ||
| Theorem | br2ndeq 36629 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵) | ||
| Theorem | dfdm5 36630 | Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
| ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | ||
| Theorem | dfrn5 36631 | Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
| ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | ||
| Theorem | opelco3 36632 | Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.) |
| ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))) | ||
| Theorem | elima4 36633 | Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.) |
| ⊢ (𝐴 ∈ (𝑅 “ 𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅) | ||
| Theorem | fv1stcnv 36634 | The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
| ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩) | ||
| Theorem | fv2ndcnv 36635 | The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩) | ||
| Theorem | elpotr 36636* | A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.) |
| ⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴) | ||
| Theorem | dford5reg 36637 | Given ax-reg 9590, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.) |
| ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) | ||
| Theorem | dfon2lem1 36638 | Lemma for dfon2 36647. (Contributed by Scott Fenton, 28-Feb-2011.) (Proof shortened by SN, 5-Oct-2026.) |
| ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} | ||
| Theorem | dfon2lem2 36639* | Lemma for dfon2 36647. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 | ||
| Theorem | dfon2lem3 36640* | Lemma for dfon2 36647. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))) | ||
| Theorem | dfon2lem4 36641* | Lemma for dfon2 36647. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
| Theorem | dfon2lem5 36642* | Lemma for dfon2 36647. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
| Theorem | dfon2lem6 36643* | Lemma for dfon2 36647. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ ((Tr 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑧((𝑧 ⊊ 𝑥 ∧ Tr 𝑧) → 𝑧 ∈ 𝑥)) → ∀𝑦((𝑦 ⊊ 𝑆 ∧ Tr 𝑦) → 𝑦 ∈ 𝑆)) | ||
| Theorem | dfon2lem7 36644* | Lemma for dfon2 36647. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵))) | ||
| Theorem | dfon2lem8 36645* | Lemma for dfon2 36647. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.) |
| ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)) → (∀𝑧((𝑧 ⊊ ∩ 𝐴 ∧ Tr 𝑧) → 𝑧 ∈ ∩ 𝐴) ∧ ∩ 𝐴 ∈ 𝐴)) | ||
| Theorem | dfon2lem9 36646* | Lemma for dfon2 36647. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E Fr 𝐴) | ||
| Theorem | dfon2 36647* | On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.) |
| ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} | ||
| Theorem | rdgprc0 36648 | The value of the recursive definition generator at ∅ when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅) | ||
| Theorem | rdgprc 36649 | The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅)) | ||
| Theorem | dfrdg2 36650* | Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}) | ||
| Theorem | dfrdg3 36651* | Generalization of dfrdg2 36650 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} | ||
| Theorem | axextdfeq 36652 | A version of ax-ext 2732 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))) | ||
| Theorem | ax8dfeq 36653 | A version of ax-8 2146 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)) | ||
| Theorem | axextdist 36654 | ax-ext 2732 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)) | ||
| Theorem | axextbdist 36655 | axextb 2735 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) | ||
| Theorem | 19.12b 36656* | Version of 19.12vv 2376 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) | ||
| Theorem | exnel 36657 | There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 | ||
| Theorem | distel 36658 | Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5405 and elirrv 9595.) (Contributed by Scott Fenton, 15-Dec-2010.) |
| ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦) | ||
| Theorem | axextndbi 36659 | axextnd 10680 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.) |
| ⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
| Theorem | hbntg 36660 | A more general form of hbnt 2327. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) | ||
| Theorem | hbimtg 36661 | A more general and closed form of hbim 2332. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃))) | ||
| Theorem | hbaltg 36662 | A more general and closed form of hbal 2203. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓)) | ||
| Theorem | hbng 36663 | A more general form of hbn 2328. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑) | ||
| Theorem | hbimg 36664 | A more general form of hbim 2332. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜃) ⇒ ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃)) | ||
| Syntax | cwsuc 36665 | Declare the syntax for well-founded successor. |
| class wsuc(𝑅, 𝐴, 𝑋) | ||
| Syntax | cwlim 36666 | Declare the syntax for well-founded limit class. |
| class WLim(𝑅, 𝐴) | ||
| Definition | df-wsuc 36667 | Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | ||
| Definition | df-wlim 36668* | Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} | ||
| Theorem | wsuceq123 36669 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌)) | ||
| Theorem | wsuceq1 36670 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋)) | ||
| Theorem | wsuceq2 36671 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋)) | ||
| Theorem | wsuceq3 36672 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) | ||
| Theorem | nfwsuc 36673 | Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑋 ⇒ ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋) | ||
| Theorem | wlimeq12 36674 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵)) | ||
| Theorem | wlimeq1 36675 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) |
| ⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴)) | ||
| Theorem | wlimeq2 36676 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) |
| ⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵)) | ||
| Theorem | nfwlim 36677 | Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) | ||
| Theorem | elwlim 36678 | Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))) | ||
| Theorem | wzel 36679 | The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴) | ||
| Theorem | wsuclem 36680* | Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))) | ||
| Theorem | wsucex 36681 | Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) | ||
| Theorem | wsuccl 36682* | If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) ⇒ ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) | ||
| Theorem | wsuclb 36683 | A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
| ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑅𝑌) ⇒ ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) | ||
| Theorem | wlimss 36684 | The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.) |
| ⊢ WLim(𝑅, 𝐴) ⊆ 𝐴 | ||
| Syntax | ctxp 36685 | Declare the syntax for tail Cartesian product. |
| class (𝐴 ⊗ 𝐵) | ||
| Syntax | cpprod 36686 | Declare the syntax for the parallel product. |
| class pprod(𝑅, 𝑆) | ||
| Syntax | csset 36687 | Declare the subset relationship class. |
| class SSet | ||
| Syntax | ctrans 36688 | Declare the transitive set class. |
| class Trans | ||
| Syntax | cbigcup 36689 | Declare the set union relationship. |
| class Bigcup | ||
| Syntax | cfix 36690 | Declare the syntax for the fixpoints of a class. |
| class Fix 𝐴 | ||
| Syntax | climits 36691 | Declare the class of limit ordinals. |
| class Limits | ||
| Syntax | cfuns 36692 | Declare the syntax for the class of all function. |
| class Funs | ||
| Syntax | csingle 36693 | Declare the syntax for the singleton function. |
| class Singleton | ||
| Syntax | csingles 36694 | Declare the syntax for the class of all singletons. |
| class Singletons | ||
| Syntax | cimage 36695 | Declare the syntax for the image functor. |
| class Image𝐴 | ||
| Syntax | ccart 36696 | Declare the syntax for the cartesian function. |
| class Cart | ||
| Syntax | cimg 36697 | Declare the syntax for the image function. |
| class Img | ||
| Syntax | cdomain 36698 | Declare the syntax for the domain function. |
| class Domain | ||
| Syntax | crange 36699 | Declare the syntax for the range function. |
| class Range | ||
| Syntax | capply 36700 | Declare the syntax for the application function. |
| class Apply | ||
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