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Theorem offveq 7425
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 4181 . . 3 (𝐴𝐴) = 𝐴
51, 2, 3, 3, 4offn 7415 . 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 7413 . . 3 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10 offveq.7 . . 3 ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))
119, 10eqtrd 2859 . 2 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐻𝑥))
125, 6, 11eqfnfvd 6797 1 (𝜑 → (𝐹f 𝑅𝐺) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115   Fn wfn 6339  cfv 6344  (class class class)co 7150  f cof 7402
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404
This theorem is referenced by:  caofid0l  7432  caofid0r  7433  caofid1  7434  caofid2  7435  ofnegsub  11635  bddibl  24449  dvaddf  24551  plydivlem3  24897  poimirlem5  35008  poimirlem10  35013  poimirlem22  35025  ofsubid  40949  ofmul12  40950  ofdivrec  40951  ofdivcan4  40952  ofdivdiv2  40953
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