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Mirrors > Home > ILE Home > Th. List > ofc1g | GIF version |
Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
ofc1.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofc1.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofc1.3 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
ofc1.4 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) |
ofc1g.ex | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐵𝑅𝐶) ∈ 𝑈) |
Ref | Expression |
---|---|
ofc1g | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofc1.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | fnconstg 5443 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
4 | ofc1.3 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | ofc1.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | inidm 3368 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | fvconst2g 5764 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
8 | 1, 7 | sylan 283 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
9 | ofc1.4 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
10 | ofc1g.ex | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐵𝑅𝐶) ∈ 𝑈) | |
11 | 3, 4, 5, 5, 6, 8, 9, 10 | ofvalg 6132 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 {csn 3618 × cxp 4653 Fn wfn 5241 ‘cfv 5246 (class class class)co 5910 ∘𝑓 cof 6120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-setind 4565 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-ov 5913 df-oprab 5914 df-mpo 5915 df-of 6122 |
This theorem is referenced by: ofnegsub 8971 |
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