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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 7987. (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 7987 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V | 
| 4 | 1, 3 | abrexex2 7994 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-uni 4908 df-iun 4993 | 
| This theorem is referenced by: plyval 26232 precsexlem4 28234 precsexlem5 28235 onmulscl 28287 zs12ex 28422 pstmfval 33895 pstmxmet 33896 | 
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