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Mirrors > Home > MPE Home > Th. List > ab2rexex | Structured version Visualization version GIF version |
Description: Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7945. (Contributed by NM, 20-Sep-2011.) |
Ref | Expression |
---|---|
ab2rexex.1 | ⊢ 𝐴 ∈ V |
ab2rexex.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
ab2rexex | ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ab2rexex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ab2rexex.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | abrexex 7945 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
4 | 1, 3 | abrexex2 7952 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2703 ∃wrex 3064 Vcvv 3468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-v 3470 df-in 3950 df-ss 3960 df-uni 4903 df-iun 4992 |
This theorem is referenced by: plyval 26077 precsexlem4 28058 precsexlem5 28059 pstmfval 33405 pstmxmet 33406 |
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