![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > caofcom | GIF version |
Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
caofref.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
caofcom.3 | ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) |
caofcom.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
caofcom | ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffvelrnda 5563 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofcom.3 | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) | |
4 | 3 | ffvelrnda 5563 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) |
5 | 2, 4 | jca 304 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) |
6 | caofcom.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) | |
7 | 6 | caovcomg 5934 | . . . 4 ⊢ ((𝜑 ∧ ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 5, 7 | syldan 280 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 8 | mpteq2dva 4026 | . 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) |
10 | caofref.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
11 | 1 | feqmptd 5482 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) |
12 | 3 | feqmptd 5482 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
13 | 10, 2, 4, 11, 12 | offval2 6005 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))) |
14 | 10, 4, 2, 12, 11 | offval2 6005 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) |
15 | 9, 13, 14 | 3eqtr4d 2183 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ↦ cmpt 3997 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ∘𝑓 cof 5988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |