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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 5477 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ↾ 𝑉) ∈ V) | |
2 | reseq1 5422 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝑉) = (𝑋 ↾ 𝑉)) | |
3 | fvtresfn.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) | |
4 | 2, 3 | fvmptg 6319 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 ↾ 𝑉) ∈ V) → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) |
5 | 1, 4 | mpdan 703 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ↦ cmpt 4762 ↾ cres 5145 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-res 5155 df-iota 5889 df-fun 5928 df-fv 5934 |
This theorem is referenced by: symgfixf1 17903 symgfixfo 17905 pwssplit1 19107 pwssplit2 19108 pwssplit3 19109 eulerpartgbij 30562 pwssplit4 37976 |
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