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Mirrors > Home > MPE Home > Th. List > abrexexg | Structured version Visualization version GIF version |
Description: Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5294, axrep6 5293, ax-rep 5286. See also abrexex2g 7951. There are partial converses under additional conditions, see for instance abnexg 7743. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2138, ax-11 2155, ax-12 2172, ax-pr 5428, ax-un 7725 and shorten proof. (Revised by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
abrexexg | ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3704 | . . 3 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵 |
3 | axrep6g 5294 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | |
4 | 2, 3 | mpan2 690 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∃*wmo 2533 {cab 2710 ∃wrex 3071 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-rep 5286 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-v 3477 |
This theorem is referenced by: abrexex 7949 iunexg 7950 qsexg 8769 wdomd 9576 cardiun 9977 rankcf 10772 sigaclci 33130 satf0suclem 34366 hbtlem1 41865 hbtlem7 41867 setpreimafvex 46051 fundcmpsurinj 46077 fundcmpsurbijinj 46078 |
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