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| Mirrors > Home > MPE Home > Th. List > ofc12 | Structured version Visualization version GIF version | ||
| Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| Ref | Expression |
|---|---|
| ofc12.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| ofc12.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| ofc12.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| Ref | Expression |
|---|---|
| ofc12 | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ofc12.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | ofc12.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
| 4 | ofc12.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) |
| 6 | fconstmpt 5747 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 8 | fconstmpt 5747 | . . . 4 ⊢ (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
| 10 | 1, 3, 5, 7, 9 | offval2 7717 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
| 11 | fconstmpt 5747 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
| 12 | 10, 11 | eqtr4di 2795 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 ↦ cmpt 5225 × cxp 5683 (class class class)co 7431 ∘f cof 7695 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
| This theorem is referenced by: pwsdiagmhm 18844 pwsdiaglmhm 21056 psrlmod 21980 coe1mul2 22272 itg2mulc 25782 dgrmulc 26311 lflvsdi2a 39081 lflvsass 39082 lflsc0N 39084 mendlmod 43201 expgrowth 44354 |
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