![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5255, axrep6 5254, ax-rep 5247. See also abrexex2g 7902. There are partial converses under additional conditions, see for instance abnexg 7695. (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 5389, ax-un 7677 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 3670 | . . 3 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵 |
3 | axrep6g 5255 | . 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 2537 {cab 2714 ∃wrex 3074 Vcvv 3448 |
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 2708 ax-rep 5247 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-v 3450 |
This theorem is referenced by: abrexex 7900 iunexg 7901 qsexg 8721 wdomd 9524 cardiun 9925 rankcf 10720 sigaclci 32771 satf0suclem 34009 hbtlem1 41479 hbtlem7 41481 setpreimafvex 45649 fundcmpsurinj 45675 fundcmpsurbijinj 45676 |
Copyright terms: Public domain | W3C validator |