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Theorem offveq 7646
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 4183 . . 3 (𝐴𝐴) = 𝐴
51, 2, 3, 3, 4offn 7635 . 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 7633 . . 3 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10 offveq.7 . . 3 ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))
119, 10eqtrd 2777 . 2 ((𝜑𝑥𝐴) → ((𝐹f 𝑅𝐺)‘𝑥) = (𝐻𝑥))
125, 6, 11eqfnfvd 6990 1 (𝜑 → (𝐹f 𝑅𝐺) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   Fn wfn 6496  cfv 6501  (class class class)co 7362  f cof 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  caofid0l  7653  caofid0r  7654  caofid1  7655  caofid2  7656  ofnegsub  12158  bddibl  25220  dvaddf  25322  plydivlem3  25671  poimirlem5  36112  poimirlem10  36117  poimirlem22  36129  fsuppssind  40797  ofsubid  42678  ofmul12  42679  ofdivrec  42680  ofdivcan4  42681  ofdivdiv2  42682
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