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Theorem caofcom 6882
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 6315 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 6315 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
52, 4jca 554 . . . 4 ((𝜑𝑤𝐴) → ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆))
6 caofcom.4 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
76caovcomg 6782 . . . 4 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆)) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
85, 7syldan 487 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
98mpteq2dva 4704 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
10 caofref.1 . . 3 (𝜑𝐴𝑉)
111feqmptd 6206 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
123feqmptd 6206 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1310, 2, 4, 11, 12offval2 6867 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
1410, 4, 2, 12, 11offval2 6867 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
159, 13, 143eqtr4d 2665 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850
This theorem is referenced by:  plydivlem4  23955  quotcan  23968  dchrabl  24879  plymulx0  30401  lfladdcom  33836  expgrowth  38013  amgmwlem  41848
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