Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isf34lem1 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-4 9204. (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 | elpw2g 4827 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
2 | 1 | biimpar 502 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴) |
3 | difexg 4808 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑋) ∈ V) | |
4 | 3 | adantr 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∖ 𝑋) ∈ V) |
5 | difeq2 3722 | . . 3 ⊢ (𝑎 = 𝑋 → (𝐴 ∖ 𝑎) = (𝐴 ∖ 𝑋)) | |
6 | compss.a | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) | |
7 | difeq2 3722 | . . . . 5 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎)) | |
8 | 7 | cbvmptv 4750 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎)) |
9 | 6, 8 | eqtri 2644 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎)) |
10 | 5, 9 | fvmptg 6280 | . 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑋) ∈ V) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) |
11 | 2, 4, 10 | syl2anc 693 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 𝒫 cpw 4158 ↦ cmpt 4729 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: compssiso 9196 isf34lem4 9199 isf34lem7 9201 isf34lem6 9202 |
Copyright terms: Public domain | W3C validator |