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Theorem caofcom 5878
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 5434 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 5434 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
52, 4jca 300 . . . 4 ((𝜑𝑤𝐴) → ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆))
6 caofcom.4 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
76caovcomg 5800 . . . 4 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆)) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
85, 7syldan 276 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
98mpteq2dva 3928 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
10 caofref.1 . . 3 (𝜑𝐴𝑉)
111feqmptd 5357 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
123feqmptd 5357 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1310, 2, 4, 11, 12offval2 5870 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
1410, 4, 2, 12, 11offval2 5870 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
159, 13, 143eqtr4d 2130 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cmpt 3899  wf 5011  cfv 5015  (class class class)co 5652  𝑓 cof 5854
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-of 5856
This theorem is referenced by: (None)
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