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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 5085 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | fconstmpt 5085 | . . . 4 ⊢ (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
10 | 1, 3, 5, 7, 9 | offval2 6812 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
11 | fconstmpt 5085 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
12 | 10, 11 | syl6eqr 2662 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 ↦ cmpt 4643 × cxp 5036 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: pwsdiagmhm 17192 pwsdiaglmhm 18878 psrlmod 19222 coe1mul2 19460 itg2mulc 23320 dgrmulc 23831 lflvsdi2a 33385 lflvsass 33386 lflsc0N 33388 mendlmod 36782 expgrowth 37556 |
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