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Theorem offveq 7696
Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
offveq.1 (𝜑𝐴𝑉)
offveq.2 (𝜑𝐹 Fn 𝐴)
offveq.3 (𝜑𝐺 Fn 𝐴)
offveq.4 (𝜑𝐻 Fn 𝐴)
offveq.5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
offveq.6 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
offveq.7 ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))
Assertion
Ref Expression
offveq (𝜑 → (𝐹f 𝑅𝐺) = 𝐻)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem offveq
StepHypRef Expression
1 offveq.2 . . 3 (𝜑𝐹 Fn 𝐴)
2 offveq.3 . . 3 (𝜑𝐺 Fn 𝐴)
3 offveq.1 . . 3 (𝜑𝐴𝑉)
4 inidm 4218 . . 3 (𝐴𝐴) = 𝐴
51, 2, 3, 3, 4offn 7685 . 2 (𝜑 → (𝐹f 𝑅𝐺) Fn 𝐴)
6 offveq.4 . 2 (𝜑𝐻 Fn 𝐴)
7 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
8 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
91, 2, 3, 3, 4, 7, 8ofval 7683 . . 3 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10 offveq.7 . . 3 ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))
119, 10eqtrd 2772 . 2 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐻𝑥))
125, 6, 11eqfnfvd 7035 1 (𝜑 → (𝐹f 𝑅𝐺) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106   Fn wfn 6538  cfv 6543  (class class class)co 7411  f cof 7670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672
This theorem is referenced by:  caofid0l  7703  caofid0r  7704  caofid1  7705  caofid2  7706  ofnegsub  12214  bddibl  25581  dvaddf  25683  plydivlem3  26032  poimirlem5  36796  poimirlem10  36801  poimirlem22  36813  fsuppssind  41467  ofsubid  43385  ofmul12  43386  ofdivrec  43387  ofdivcan4  43388  ofdivdiv2  43389
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