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Mirrors > Home > MPE Home > Th. List > fvtresfn | Structured version Visualization version GIF version |
Description: Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
fvtresfn.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) |
Ref | Expression |
---|---|
fvtresfn | ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexg 5654 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ↾ 𝑉) ∈ V) | |
2 | reseq1 5594 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝑉) = (𝑋 ↾ 𝑉)) | |
3 | fvtresfn.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) | |
4 | 2, 3 | fvmptg 6505 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 ↾ 𝑉) ∈ V) → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) |
5 | 1, 4 | mpdan 679 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ↦ cmpt 4922 ↾ cres 5314 ‘cfv 6101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-res 5324 df-iota 6064 df-fun 6103 df-fv 6109 |
This theorem is referenced by: symgfixf1 18169 symgfixfo 18171 pwssplit1 19380 pwssplit2 19381 pwssplit3 19382 eulerpartgbij 30950 pwssplit4 38444 |
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