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| Mirrors > Home > MPE Home > Th. List > fabex | Structured version Visualization version GIF version | ||
| Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007.) | 
| Ref | Expression | 
|---|---|
| fabex.1 | ⊢ 𝐴 ∈ V | 
| fabex.2 | ⊢ 𝐵 ∈ V | 
| fabex.3 | ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} | 
| Ref | Expression | 
|---|---|
| fabex | ⊢ 𝐹 ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fabex.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | fabex.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | fabex.3 | . . 3 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} | |
| 4 | 3 | fabexg 7960 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | 
| 5 | 1, 2, 4 | mp2an 692 | 1 ⊢ 𝐹 ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⟶wf 6557 | 
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: (None) | 
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