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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 7972. (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 7972 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
4 | 1, 3 | abrexex2 7979 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2705 ∃wrex 3067 Vcvv 3473 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-v 3475 df-in 3956 df-ss 3966 df-uni 4913 df-iun 5002 |
This theorem is referenced by: plyval 26147 precsexlem4 28128 precsexlem5 28129 pstmfval 33530 pstmxmet 33531 |
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