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Mirrors > Home > MPE Home > Th. List > isf34lem2 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-4 10399. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
compss.a | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
Ref | Expression |
---|---|
isf34lem2 | ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4127 | . . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 | |
2 | elpw2g 5340 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 258 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴) |
5 | compss.a | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) | |
6 | 4, 5 | fmptd 7118 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∖ cdif 3942 ⊆ wss 3945 𝒫 cpw 4598 ↦ cmpt 5225 ⟶wf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6544 df-fn 6545 df-f 6546 |
This theorem is referenced by: isf34lem5 10395 isf34lem7 10396 isf34lem6 10397 |
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