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Mirrors > Home > MPE Home > Th. List > ofc2 | Structured version Visualization version GIF version |
Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014.) |
Ref | Expression |
---|---|
ofc2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofc2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofc2.3 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
ofc2.4 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) |
Ref | Expression |
---|---|
ofc2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofc2.3 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | ofc2.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | fnconstg 6780 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
5 | ofc2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | inidm 4215 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | ofc2.4 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
8 | fvconst2g 7209 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
9 | 2, 8 | sylan 579 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
10 | 1, 4, 5, 5, 6, 7, 9 | ofval 7691 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4625 × cxp 5671 Fn wfn 6538 ‘cfv 6543 (class class class)co 7415 ∘f cof 7678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 |
This theorem is referenced by: lflvscl 38544 lkrsc 38564 ldualvsval 38605 |
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