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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 7896. (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 7896 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
4 | 1, 3 | abrexex2 7903 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2714 ∃wrex 3074 Vcvv 3446 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-v 3448 df-in 3918 df-ss 3928 df-uni 4867 df-iun 4957 |
This theorem is referenced by: plyval 25557 pstmfval 32480 pstmxmet 32481 |
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