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| Mirrors > Home > MPE Home > Th. List > caofid0l | Structured version Visualization version GIF version | ||
| Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| caofref.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| caofref.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | 
| caofid0.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| caofid0l.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥) | 
| Ref | Expression | 
|---|---|
| caofid0l | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | caofref.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | caofid0.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | fnconstg 6796 | . . 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 6737 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| 7 | fvconst2g 7222 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
| 8 | 2, 7 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | 
| 9 | eqidd 2738 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
| 10 | caofid0l.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥) | |
| 11 | 10 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥) | 
| 12 | 5 | ffvelcdmda 7104 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) | 
| 13 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤))) | |
| 14 | id 22 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
| 15 | 13, 14 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤))) | 
| 16 | 15 | rspccva 3621 | . . 3 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)) | 
| 17 | 11, 12, 16 | syl2an2r 685 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)) | 
| 18 | 1, 4, 6, 6, 8, 9, 17 | offveq 7723 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 | 
| This theorem is referenced by: mndvlid 18812 psr0lid 21973 psrlmod 21980 lfladd0l 39075 mendlmod 43201 | 
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