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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 6735 | . . 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 6674 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | fvconst2g 7156 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵) | |
8 | 2, 7 | sylan 581 | . 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 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥) |
12 | 5 | ffvelcdmda 7040 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
13 | oveq2 7370 | . . . . 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 3583 | . . 3 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)) |
17 | 11, 12, 16 | syl2an2r 684 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)) |
18 | 1, 4, 6, 6, 8, 9, 17 | offveq 7646 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 {csn 4591 × cxp 5636 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: psr0lid 21379 psrlmod 21386 mndvlid 21758 lfladd0l 37565 mendlmod 41549 |
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