MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-rep Structured version   Visualization version   GIF version

Axiom ax-rep 4693
Description: Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 5876). Although 𝜑 may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and 𝜑 encodes the predicate "the value of the function at 𝑤 is 𝑧." Thus, 𝜑 will ordinarily have free variables 𝑤 and 𝑧- think of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the quantifier 𝑦 in order to "protect" the axiom from any 𝜑 containing 𝑦, thus allowing us to eliminate any restrictions on 𝜑. Another common variant is derived as axrep5 4698, where you can find some further remarks. A slightly more compact version is shown as axrep2 4695. A quite different variant is zfrep6 7004, which if used in place of ax-rep 4693 would also require that the Separation Scheme axsep 4702 be stated as a separate axiom.

There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 8614 and the Boundedness Axiom bnd 8615.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4702, Null Set axnul 4710, and Pairing axpr 4827, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 4703, ax-nul 4712, and ax-pr 4828 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-rep (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Detailed syntax breakdown of Axiom ax-rep
StepHypRef Expression
1 wph . . . . . . 7 wff 𝜑
2 vy . . . . . . 7 setvar 𝑦
31, 2wal 1472 . . . . . 6 wff 𝑦𝜑
4 vz . . . . . . 7 setvar 𝑧
54, 2weq 1860 . . . . . 6 wff 𝑧 = 𝑦
63, 5wi 4 . . . . 5 wff (∀𝑦𝜑𝑧 = 𝑦)
76, 4wal 1472 . . . 4 wff 𝑧(∀𝑦𝜑𝑧 = 𝑦)
87, 2wex 1694 . . 3 wff 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
9 vw . . 3 setvar 𝑤
108, 9wal 1472 . 2 wff 𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
114, 2wel 1977 . . . . 5 wff 𝑧𝑦
12 vx . . . . . . . 8 setvar 𝑥
139, 12wel 1977 . . . . . . 7 wff 𝑤𝑥
1413, 3wa 382 . . . . . 6 wff (𝑤𝑥 ∧ ∀𝑦𝜑)
1514, 9wex 1694 . . . . 5 wff 𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)
1611, 15wb 194 . . . 4 wff (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1716, 4wal 1472 . . 3 wff 𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1817, 2wex 1694 . 2 wff 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1910, 18wi 4 1 wff (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
This axiom is referenced by:  axrep1  4694  axnulALT  4709  bj-axrep1  31782  bj-snsetex  31940
  Copyright terms: Public domain W3C validator