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Theorem caofcom 5676
 Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (φA 𝑉)
caofref.2 (φ𝐹:A𝑆)
caofcom.3 (φ𝐺:A𝑆)
caofcom.4 ((φ (x 𝑆 y 𝑆)) → (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 5245 . . . . 5 ((φ w A) → (𝐹w) 𝑆)
3 caofcom.3 . . . . . 6 (φ𝐺:A𝑆)
43ffvelrnda 5245 . . . . 5 ((φ w A) → (𝐺w) 𝑆)
52, 4jca 290 . . . 4 ((φ w A) → ((𝐹w) 𝑆 (𝐺w) 𝑆))
6 caofcom.4 . . . . 5 ((φ (x 𝑆 y 𝑆)) → (x𝑅y) = (y𝑅x))
76caovcomg 5598 . . . 4 ((φ ((𝐹w) 𝑆 (𝐺w) 𝑆)) → ((𝐹w)𝑅(𝐺w)) = ((𝐺w)𝑅(𝐹w)))
85, 7syldan 266 . . 3 ((φ w A) → ((𝐹w)𝑅(𝐺w)) = ((𝐺w)𝑅(𝐹w)))
98mpteq2dva 3838 . 2 (φ → (w A ↦ ((𝐹w)𝑅(𝐺w))) = (w A ↦ ((𝐺w)𝑅(𝐹w))))
10 caofref.1 . . 3 (φA 𝑉)
111feqmptd 5169 . . 3 (φ𝐹 = (w A ↦ (𝐹w)))
123feqmptd 5169 . . 3 (φ𝐺 = (w A ↦ (𝐺w)))
1310, 2, 4, 11, 12offval2 5668 . 2 (φ → (𝐹𝑓 𝑅𝐺) = (w A ↦ ((𝐹w)𝑅(𝐺w))))
1410, 4, 2, 12, 11offval2 5668 . 2 (φ → (𝐺𝑓 𝑅𝐹) = (w A ↦ ((𝐺w)𝑅(𝐹w))))
159, 13, 143eqtr4d 2079 1 (φ → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ↦ cmpt 3809  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455   ∘𝑓 cof 5652 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654 This theorem is referenced by: (None)
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