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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 6566 | . . 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 6514 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | fvconst2g 6963 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
8 | 2, 7 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) |
9 | eqidd 2822 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
10 | caofid0l.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥) | |
11 | 10 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥) |
12 | 5 | ffvelrnda 6850 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
13 | oveq2 7163 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤))) | |
14 | id 22 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
15 | 13, 14 | eqeq12d 2837 | . . . 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 683 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)) |
18 | 1, 4, 6, 6, 8, 9, 17 | offveq 7429 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {csn 4566 × cxp 5552 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∘f cof 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 |
This theorem is referenced by: psr0lid 20174 psrlmod 20180 mndvlid 21003 lfladd0l 36209 mendlmod 39791 |
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