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Mirrors > Home > MPE Home > Th. List > caofid1 | 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) |
caofid1.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶) |
Ref | Expression |
---|---|
caofid1 | ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | caofref.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
3 | 2 | ffnd 6711 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | caofid0.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | fnconstg 6772 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
7 | caofid1.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | fnconstg 6772 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) |
10 | eqidd 2727 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
11 | fvconst2g 7198 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
12 | 4, 11 | sylan 579 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) |
13 | caofid1.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶) | |
14 | 13 | ralrimiva 3140 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶) |
15 | 2 | ffvelcdmda 7079 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
16 | oveq1 7411 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵)) | |
17 | 16 | eqeq1d 2728 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹‘𝑤)𝑅𝐵) = 𝐶)) |
18 | 17 | rspccva 3605 | . . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶) |
19 | 14, 15, 18 | syl2an2r 682 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶) |
20 | fvconst2g 7198 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) | |
21 | 7, 20 | sylan 579 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) |
22 | 19, 21 | eqtr4d 2769 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤)) |
23 | 1, 3, 6, 9, 10, 12, 22 | offveq 7690 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {csn 4623 × cxp 5667 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∘f cof 7664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 |
This theorem is referenced by: plymul0or 26165 fta1lem 26192 lfl0sc 38464 |
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