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Theorem List for Metamath Proof Explorer - 39801-39900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmdandyvrx4 39801 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx5 39802 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx6 39803 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx7 39804 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx8 39805 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx9 39806 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx10 39807 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx11 39808 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx12 39809 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx13 39810 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx14 39811 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx15 39812 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

TheoremH15NH16TH15IH16 39813 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   𝜏    &   𝜂    &   𝜁    &   𝜎    &   𝜌    &   𝜇    &   𝜆    &   𝜅    &   jph    &   jps    &   jch    &   jth       (((((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ jph) ∧ jps) ∧ jch) → jth)

Theoremdandysum2p2e4 39814

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

(𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)    &   (𝜁 ↔ ⊤)    &   (𝜎 ↔ ⊥)    &   (𝜌 ↔ ⊥)    &   (𝜇 ↔ ⊥)    &   (𝜆 ↔ ⊥)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Theoremmdandysum2p2e4 39815 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half adder and full adder in propositional calculus.

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

(jth ↔ ⊥)    &   (jta ↔ ⊤)    &   (𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃jth)    &   (𝜏jth)    &   (𝜂jta)    &   (𝜁jta)    &   (𝜎jth)    &   (𝜌jth)    &   (𝜇jth)    &   (𝜆jth)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

21.34  Mathbox for Alexander van der Vekens

21.34.1  Double restricted existential uniqueness

21.34.1.1  Restricted quantification (extension)

Theoremr19.32 39816 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3064. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))

Theoremrexsb 39817* An equivalent expression for restricted existence, analogous to exsb 2456. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))

Theoremrexrsb 39818* An equivalent expression for restricted existence, analogous to exsb 2456. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))

Theorem2rexsb 39819* An equivalent expression for double restricted existence, analogous to rexsb 39817. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theorem2rexrsb 39820* An equivalent expression for double restricted existence, analogous to 2exsb 2457. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremcbvral2 39821* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3155. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2 39822* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3156. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theorem2ralbiim 39823 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1807 and ralbiim 3051. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))

21.34.1.2  The empty set (extension)

Theoremraaan2 39824* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4032. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑦𝜑    &   𝑥𝜓       ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))

21.34.1.3  Restricted uniqueness and "at most one" quantification

Theoremrmoimi 39825 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)

Theorem2reu5a 39826 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))

Theoremreuimrmo 39827 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2510. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Theoremrmoanim 39828* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2517. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
𝑥𝜑       (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))

Theoremreuan 39829* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2518. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑥𝜑       (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))

21.34.1.4  Analogs to Existential uniqueness (double quantification)

Theorem2reurex 39830* Double restricted quantification with existential uniqueness, analogous to 2euex 2532. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)

Theorem2reurmo 39831* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2533. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu2rex 39832* Double restricted existential uniqueness, analogous to 2eu2ex 2534. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theorem2rmoswap 39833* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2535. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))

Theorem2rexreu 39834* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2537. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu1 39835* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2541. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu2 39836* Double restricted existential uniqueness, analogous to 2eu2 2542. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))

Theorem2reu3 39837* Double restricted existential uniqueness, analogous to 2eu3 2543. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∀𝑥𝐴𝑦𝐵 (∃*𝑥𝐴 𝜑 ∨ ∃*𝑦𝐵 𝜑) → ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu4a 39838* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2544 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 39839). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))

Theorem2reu4 39839* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2544. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2reu7 39840* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2547. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

Theorem2reu8 39841* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2548. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 39840. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

21.34.2  Alternative definitions of function's and operation's values

The current definition of the value (𝐹𝐴) of a function 𝐹 for an argument 𝐴 (see df-fv 5812) assures that this value is always a set, see fex 6394. This is because this definition can be applied to any classes 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6128 and fvprc 6097).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹𝐴) = ∅ alone it cannot be decided/derived if (𝐹𝐴) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value ) or not. Therefore, additional assumptions are required, such as ∅ ∉ ran 𝐹, ∅ ∈ ran 𝐹 or Fun 𝐹𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6129).

To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 39846) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 39868, ndmafv 39869, afvprc 39873 and nfunsnafv 39871), and which corresponds to the current definition ((𝐹𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 39867). That means (𝐹'''𝐴) = V → (𝐹𝐴) = ∅ (see afvpcfv0 39875), but (𝐹𝐴) = ∅ → (𝐹'''𝐴) = V is not generally valid.

In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴, which also would result in (𝐹'''𝐴) = V. In this context we say (𝐹'''𝐴) "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds: (𝐹'''𝐴) ∈ V <-> "(𝐹'''𝐴) is meaningful/defined".

An interesting question would be if (𝐹𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function's values. If we look at the (currently 19) proofs using the definition df-fv 5812 of (𝐹𝐴), we see that analogons for the following 8 theorems can be proven using the alternative definition: fveq1 6102-> afveq1 39863, fveq2 6103-> afveq2 39864, nffv 6110-> nfafv 39865, csbfv12 6141-> csbafv12g , fvres 6117-> afvres 39901, rlimdm 14130-> rlimdmafv 39906, tz6.12-1 6120-> tz6.12-1-afv 39903, fveu 6095-> afveu 39882.

3 theorems proved by directly using df-fv 5812 are within a mathbox (fvsb 37677) or not used (isumclim3 14332, avril1 26711).

However, the remaining 8 theorems proved by directly using df-fv 5812 are used more or less often:

* fvex 6113: used in about 1750 proofs.

* tz6.12-1 6120: root theorem of many theorems which have not a strict analogon, and which are used many times: fvprc 6097 (used in about 127 proofs), tz6.12i 6124 (used - indirectly via fvbr0 6125 and fvrn0 6126- in 18 proofs, and in fvclss 6404 used in fvclex 7031 used in fvresex 7032, which is not used!), dcomex 9152 (used in 4 proofs), ndmfv 6128 (used in 86 proofs) and nfunsn 6135 (used by dffv2 6181 which is not used).

* fv2 6098: only used by elfv 6101, which is only used by fv3 6116, which is not used.

* dffv3 6099: used by dffv4 6100 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTOLD 38074, csbfv12gALTVD 38157), by shftval 13662 (itself used in 9 proofs), by dffv5 31201 (mathbox) and by fvco2 6183, which has the analogon afvco2 39905.

* fvopab5 6217: used only by ajval 27101 (not used) and by adjval 28133 ( used - indirectly - in 9 proofs).

* zsum 14296: used (via isum 14297, sum0 14299 and fsumsers 14306) in more than 90 proofs.

* isumshft 14410: used in pserdv2 23988 and (via logtayl 24206) 4 other proofs.

* ovtpos 7254: used in 14 proofs.

As a result of this analysis we can say that the current definition of a function's value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 6098, dffv3 6099, fvopab5 6217, zsum 14296, isumshft 14410 and ovtpos 7254 are not critical or are, hopefully, also valid for the alternative definition, fvex 6113 and tz6.12-1 6120 (and the theorems based on them) are essential for the current definition of function values.

With the same arguments, an alternatvie definition of operation's values ((𝐴𝑂𝐵)) could be meaningful to avoid ambiguities, see df-aov 39847.

Syntaxwdfat 39842 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function) 𝐹 is defined at (the argument) 𝐴). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for \$ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
wff 𝐹 defAt 𝐴

Syntaxcafv 39843 Extend the definition of a class to include the value of a function. (Read: The value of 𝐹 at 𝐴, or "𝐹 of 𝐴."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol used for the current definition of a function's value (see df-fv 5812), which, by the way, was intended to visualize that in many cases and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 39861, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5812 and df-ima 5051. And not three backticks ( three times ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
class (𝐹'''𝐴)

Syntaxcaov 39844 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by a pair of parentheses in contrast to the current definition, see df-ov 6552.
class ((𝐴𝐹𝐵))

Definitiondf-dfat 39845 Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Definitiondf-afv 39846* Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹𝐴) = ∅ (see df-fv 5812 and ndmfv 6128), (𝐹'''𝐴) = V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)

Definitiondf-aov 39847 Define the value of an operation. In contrast to df-ov 6552, the alternative definition for a function value (see df-afv 39846) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)

21.34.2.1  Restricted quantification (extension)

Theoremralbinrald 39848* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
(𝜑𝑋𝐴)    &   (𝑥𝐴𝑥 = 𝑋)    &   (𝑥 = 𝑋 → (𝜓𝜃))       (𝜑 → (∀𝑥𝐴 𝜓𝜃))

21.34.2.2  The universal class (extension)

Theoremnvelim 39849 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4725. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴 = V → ¬ 𝐴𝐵)

21.34.2.3  Introduce the Axiom of Power Sets (extension)

Theoremalneu 39850 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
(∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Theoremeu2ndop1stv 39851* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)

21.34.2.4  Relations (extension)

Theoremeldmressn 39852 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

21.34.2.5  Functions (extension)

Theoremfveqvfvv 39853 If a function's value at an argument is the universal class (which can never be the case because of fvex 6113), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 115). (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)

Theoremfunresfunco 39854 Composition of two functions, generalization of funco 5842. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))

Theoremfnresfnco 39855 Composition of two functions, similar to fnco 5913. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)

Theoremfuncoressn 39856 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))

Theoremfunressnfv 39857 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))

21.34.2.6  Predicate "defined at"

Theoremdfateq12d 39858 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))

Theoremnfdfat 39859 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 defAt 𝐴

Theoremdfdfat2 39860* Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

21.34.2.7  Alternative definition of the value of a function

Theoremdfafv2 39861 Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Theoremafveq12d 39862 Equality deduction for function value, analogous to fveq12d 6109. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 39863 Equality theorem for function value, analogous to fveq1 6102. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 39864 Equality theorem for function value, analogous to fveq1 6102. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 39865 Bound-variable hypothesis builder for function value, analogous to nffv 6110. To prove a deduction version of this analogous to nffvd 6112 is not easily possible because a deduction version of nfdfat 39859 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 39866 Move class substitution in and out of a function value, analogous to csbfv12 6141, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6585. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 39867 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvnfundmuv 39868 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Theoremndmafv 39869 The value of a class outside its domain is the universe, compare with ndmfv 6128. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 39870 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)

Theoremnfunsnafv 39871 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6135. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 39872 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))

Theoremafvprc 39873 A function's value at a proper class is the universe, compare with fvprc 6097. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 39874 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)

Theoremafvpcfv0 39875 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)

Theoremafvnufveq 39876 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvvfveq 39877 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0fv0 39878 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)

Theoremafvfvn0fveq 39879 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0nbfvbi 39880 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
(∅ ∉ 𝐵 → ((𝐹'''𝐴) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))

Theoremafvfv0bi 39881 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Theoremafveu 39882* The value of a function at a unique point, analogous to fveu 6095. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 39883 Equivalence of function value and binary relation, analogous to fnbrfvb 6146. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 39884 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6147. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 39885 Equivalence of function value and binary relation, analogous to funbrfvb 6148. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 39886 Equivalence of function value and ordered pair membership, analogous to funopfvb 6149. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Theoremfunbrafv 39887 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6144. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 39888 Function value in terms of a binary relation, analogous to funbrfv2b 6150. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 39889* Representation of a function in terms of its values, analogous to dffn5 6151 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 39890* Representation of a function in terms of its values, analogous to dffn5 6151 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
(∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))

Theoremfnrnafv 39891* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6152. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 39892* A member of a function's range is a value of the function, analogous to fvelrnb 6153 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremafvelrnb0 39893* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6153. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 39894* Alternate definition of the image of a function, analogous to dfimafn 6155. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 39895* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6156. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 39896* Function value in an image, analogous to fvelima 6158. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 39897 A function's value belongs to its range, analogous to fvelrn 6260. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 39898 A function's value belongs to its range, analogous to fnfvelrn 6264. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 39899 A function's value belongs to its codomain, analogous to ffvelrn 6265. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 39900* A function maps to a class to which all values belong, analogous to ffnfv 6295. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

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