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Mirrors > Home > MPE Home > Th. List > fabexg | Structured version Visualization version GIF version |
Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.) |
Ref | Expression |
---|---|
fabexg.1 | ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} |
Ref | Expression |
---|---|
fabexg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7733 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V) | |
2 | pwexg 5369 | . 2 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V) | |
3 | fabexg.1 | . . . . 5 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} | |
4 | fssxp 6738 | . . . . . . . 8 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ⊆ (𝐴 × 𝐵)) | |
5 | velpw 4602 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵)) | |
6 | 4, 5 | sylibr 233 | . . . . . . 7 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ∈ 𝒫 (𝐴 × 𝐵)) |
7 | 6 | anim1i 614 | . . . . . 6 ⊢ ((𝑥:𝐴⟶𝐵 ∧ 𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)) |
8 | 7 | ss2abi 4058 | . . . . 5 ⊢ {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} |
9 | 3, 8 | eqsstri 4011 | . . . 4 ⊢ 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} |
10 | ssab2 4071 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵) | |
11 | 9, 10 | sstri 3986 | . . 3 ⊢ 𝐹 ⊆ 𝒫 (𝐴 × 𝐵) |
12 | ssexg 5316 | . . 3 ⊢ ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V) | |
13 | 11, 12 | mpan 687 | . 2 ⊢ (𝒫 (𝐴 × 𝐵) ∈ V → 𝐹 ∈ V) |
14 | 1, 2, 13 | 3syl 18 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 × cxp 5667 ⟶wf 6532 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-fun 6538 df-fn 6539 df-f 6540 |
This theorem is referenced by: fabex 7922 f1oabexg 7923 elghomlem1OLD 37264 |
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