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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 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
4 | ofc12.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) |
6 | fconstmpt 5699 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | fconstmpt 5699 | . . . 4 ⊢ (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
10 | 1, 3, 5, 7, 9 | offval2 7642 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
11 | fconstmpt 5699 | . 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 1542 ∈ wcel 2107 {csn 4591 ↦ cmpt 5193 × cxp 5636 (class class class)co 7362 ∘f cof 7620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: pwsdiagmhm 18648 pwsdiaglmhm 20534 psrlmod 21386 coe1mul2 21656 itg2mulc 25128 dgrmulc 25648 lflvsdi2a 37571 lflvsass 37572 lflsc0N 37574 mendlmod 41549 expgrowth 42689 |
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