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Theorem caofcom 7733
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)
21ffvelcdmda 7103 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
43ffvelcdmda 7103 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
52, 4jca 511 . . . 4 ((𝜑𝑤𝐴) → ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆))
6 caofcom.4 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
76caovcomg 7627 . . . 4 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆)) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
85, 7syldan 591 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
98mpteq2dva 5247 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
10 caofref.1 . . 3 (𝜑𝐴𝑉)
111feqmptd 6976 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
123feqmptd 6976 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1310, 2, 4, 11, 12offval2 7716 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
1410, 4, 2, 12, 11offval2 7716 . 2 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
159, 13, 143eqtr4d 2784 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝐺f 𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696
This theorem is referenced by:  plydivlem4  26352  quotcan  26365  dchrabl  27312  plymulx0  34540  lfladdcom  39053  expgrowth  44330  amgmwlem  49032
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