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Mirrors > Home > MPE Home > Th. List > caofid2 | Structured version Visualization version GIF version |
Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
caofref.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
caofid0.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
caofid1.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
caofid2.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶) |
Ref | Expression |
---|---|
caofid2 | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | caofid0.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | fnconstg 6569 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
5 | caofref.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | 5 | ffnd 6517 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | caofid1.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | fnconstg 6569 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) |
10 | fvconst2g 6966 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
11 | 2, 10 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) |
12 | eqidd 2824 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
13 | caofid2.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶) | |
14 | 13 | ralrimiva 3184 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶) |
15 | 5 | ffvelrnda 6853 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
16 | oveq2 7166 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤))) | |
17 | 16 | eqeq1d 2825 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹‘𝑤)) = 𝐶)) |
18 | 17 | rspccva 3624 | . . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶) |
19 | 14, 15, 18 | syl2an2r 683 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶) |
20 | fvconst2g 6966 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) | |
21 | 7, 20 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) |
22 | 19, 21 | eqtr4d 2861 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = ((𝐴 × {𝐶})‘𝑤)) |
23 | 1, 4, 6, 9, 11, 12, 22 | offveq 7432 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {csn 4569 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 |
This theorem is referenced by: mbfmulc2lem 24250 i1fmulc 24306 itg1mulc 24307 itg2mulc 24350 dvcmulf 24544 coe0 24848 plymul0or 24872 0prjspnrel 39276 expgrowth 40674 |
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