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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 7633 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V) |
5 | 1, 2, 4 | mp2an 690 | 1 ⊢ 𝐹 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3494 ⟶wf 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-fun 6351 df-fn 6352 df-f 6353 |
This theorem is referenced by: (None) |
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