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Theorem offveq 6915
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 3820 . . 3 (𝐴𝐴) = 𝐴
51, 2, 3, 3, 4offn 6905 . 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 6903 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10 offveq.7 . . 3 ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))
119, 10eqtrd 2655 . 2 ((𝜑𝑥𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑥) = (𝐻𝑥))
125, 6, 11eqfnfvd 6312 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989   Fn wfn 5881  cfv 5886  (class class class)co 6647  𝑓 cof 6892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894
This theorem is referenced by:  caofid0l  6922  caofid0r  6923  caofid1  6924  caofid2  6925  ofnegsub  11015  bddibl  23600  dvaddf  23699  plydivlem3  24044  poimirlem5  33394  poimirlem10  33399  poimirlem22  33411  ofsubid  38349  ofmul12  38350  ofdivrec  38351  ofdivcan4  38352  ofdivdiv2  38353
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