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Mirrors > Home > MPE Home > Th. List > ofmresval | Structured version Visualization version GIF version |
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.) |
Ref | Expression |
---|---|
ofmresval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐴) |
ofmresval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
ofmresval | ⊢ (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘f 𝑅𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofmresval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴) | |
2 | ofmresval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
3 | ovres 7308 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐵) → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘f 𝑅𝐺)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘f 𝑅𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 × cxp 5547 ↾ cres 5551 (class class class)co 7150 ∘f cof 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-res 5561 df-iota 6308 df-fv 6357 df-ov 7153 |
This theorem is referenced by: psradd 20156 dchrmul 25818 ldualvadd 36259 |
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