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Mirrors > Home > MPE Home > Th. List > isf34lem1 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-4 10419. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
compss.a | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
Ref | Expression |
---|---|
isf34lem1 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | compss.a | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) | |
2 | difeq2 4129 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎)) | |
3 | 2 | cbvmptv 5260 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎)) |
4 | 1, 3 | eqtri 2762 | . 2 ⊢ 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎)) |
5 | difeq2 4129 | . 2 ⊢ (𝑎 = 𝑋 → (𝐴 ∖ 𝑎) = (𝐴 ∖ 𝑋)) | |
6 | elpw2g 5338 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
7 | 6 | biimpar 477 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴) |
8 | difexg 5334 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑋) ∈ V) | |
9 | 8 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∖ 𝑋) ∈ V) |
10 | 4, 5, 7, 9 | fvmptd3 7038 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 𝒫 cpw 4604 ↦ cmpt 5230 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: compssiso 10411 isf34lem4 10414 isf34lem7 10416 isf34lem6 10417 |
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