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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 6331 | . . 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 4048 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | ofc2.4 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
8 | fvconst2g 6724 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
9 | 2, 8 | sylan 577 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
10 | 1, 4, 5, 5, 6, 7, 9 | ofval 7167 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4398 × cxp 5341 Fn wfn 6119 ‘cfv 6124 (class class class)co 6906 ∘𝑓 cof 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 |
This theorem is referenced by: lflvscl 35153 lkrsc 35173 ldualvsval 35214 |
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