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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 6736 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| 4 | caofid0.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 5 | fnconstg 6795 | . . 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 6795 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴) | |
| 9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) | 
| 10 | eqidd 2737 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
| 11 | fvconst2g 7223 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
| 12 | 4, 11 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | 
| 13 | caofid1.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶) | |
| 14 | 13 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶) | 
| 15 | 2 | ffvelcdmda 7103 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) | 
| 16 | oveq1 7439 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵)) | |
| 17 | 16 | eqeq1d 2738 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹‘𝑤)𝑅𝐵) = 𝐶)) | 
| 18 | 17 | rspccva 3620 | . . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶) | 
| 19 | 14, 15, 18 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶) | 
| 20 | fvconst2g 7223 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) | |
| 21 | 7, 20 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶) | 
| 22 | 19, 21 | eqtr4d 2779 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤)) | 
| 23 | 1, 3, 6, 9, 10, 12, 22 | offveq 7724 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {csn 4625 × cxp 5682 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 | 
| This theorem is referenced by: plymul0or 26323 fta1lem 26350 lfl0sc 39084 | 
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