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Theorem caofcom 5785
Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofcom.3 (𝜑𝐺:𝐴𝑆)
caofcom.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
Assertion
Ref Expression
caofcom (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caofcom
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
21ffvelrnda 5354 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 5354 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
52, 4jca 300 . . . 4 ((𝜑𝑤𝐴) → ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆))
6 caofcom.4 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
76caovcomg 5707 . . . 4 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆)) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
85, 7syldan 276 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
98mpteq2dva 3888 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
10 caofref.1 . . 3 (𝜑𝐴𝑉)
111feqmptd 5278 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
123feqmptd 5278 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1310, 2, 4, 11, 12offval2 5777 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
1410, 4, 2, 12, 11offval2 5777 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
159, 13, 143eqtr4d 2125 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  cmpt 3859  wf 4948  cfv 4952  (class class class)co 5563  𝑓 cof 5761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-of 5763
This theorem is referenced by: (None)
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